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https://alumniagri.in/task/the-locus-of-the-mid-point-of-the-portion-of-the-line-x-cos-22631274
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Math, asked by bommusashi, 10 days ago
# The locus of the mid point of the portion of the linex cos a + y sin a = p between the coordinate axes is (p is a nonzero constant)(A) x2 + y2 = 1(B) +4TELEy2(C) x2 + y2 = 4p?(D) 3 +4
0
\bf{\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}}
x
2
1
+
y
2
1
=
p
2
4
Step-by-step explanation:
Let P(h,k) be the midpoint of the portion of the line
x\:cos\alpha+y\:sin\alpha=pxcosα+ysinα=p .....(1)
put y=0 in (1), we get
x\:cos\alpha=pxcosα=p
x=\frac{p}{cos\alpha}x=
cosα
p
put x=0 in(1), we get
y\:sin\alpha=pysinα=p
y=\frac{p}{sin\alpha}y=
sinα
p
Therefore, the line (1) meets the coordinate axes at
A(\frac{p}{cos\alpha},0
cosα
p
,0 ) and B(0,\frac{p}{sin\alpha}
sinα
p
)
Clearly, the midpoint of the Portion AB= P
(\frac{\frac{p}{cos\alpha}+0}{2},\frac{0+\frac{p}{sin\alpha}}{2})=(h,k)(
2
cosα
p
+0
,
2
0+
sinα
p
)=(h,k)
(\frac{p}{2cos\alpha},\frac{p}{2sin\alpha})=(h,k)(
2cosα
p
,
2sinα
p
)=(h,k)
\implies\:h=\frac{p}{2cos\alpha},\:\:\:\:k=\frac{p}{2sinalpha}⟹h=
2cosα
p
,k=
2sinalpha
p
\implies\:cos\alpha=\frac{p}{2h},\:\:\:\:sin\alpha=\frac{p}{2k}⟹cosα=
2h
p
,sinα=
2k
p
squaring and adding these equations, we get
cos^2\alpha+sin^2\alpha=\frac{p^2}{4h^2}+\frac{p^2}{4k^2}cos
2
α+sin
2
α=
4h
2
p
2
+
4k
2
p
2
\implies\:1=\frac{p^2}{4h^2}+\frac{p^2}{4k^2}⟹1=
4h
2
p
2
+
4k
2
p
2
\implies\:1=\frac{p^2}{4}(\frac{1}{h^2}+\frac{1}{k^2})⟹1=
4
p
2
(
h
2
1
+
k
2
1
)
\implies\:\frac{4}{p^2}=\frac{1}{h^2}+\frac{1}{k^2}⟹
p
2
4
=
h
2
1
+
k
2
1
\therefore\text{The locus of P is}∴The locus of P is
\boxed{\bf{\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}}}
x
2
1
+
y
2
1
=
p
2
4
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# Odd/even extensions
### Odd/even extensions
Good evening all,
I have this question,
f(t) is defined on the interval 0 ≤ t < 2 by f(t) = t(2 − t).
a) sketch the odd extension of f for -6≤ t ≤ 6, and state the fundamental period of this extension.
b) sketch the even extension for -6≤ t ≤ 6 and state the fundamental period of this extension.
The graph I have ended up with doesn't seem right, as a t=0, it is 0, t=1 it is 1, t=2 it is 0, and then as I plot up to t=6, i end up getting -24, is this completely wrong? should I just repeat the 0,1,0 pattern? I also can't figure out my fundamental periods.
Hope someone can help.
Thank you.
Jaffacake
Posts: 2
Joined: Sun Dec 29, 2019 6:53 am
Reputation: 0
### Re: Odd/even extensions
First, can you graph f(x)= x(2- x) for x between 0 and 2? (It is a parabola opening downward and 0 at x= 0 and x= 2 with vertex at (1, 10).)
An "even" extension just copies that piece of parabola for x= 2 to 4, x= 4 to 6, etc. as well as for x= -2 to 0, x= -4 to -2, etc. Do you see that it has period 2?
An "odd extension" flips that parabola upside down for x= 2 to 4, back upright for x= 4 to 6, upside down again for x= 6 to 8, etc. And "upside down" for x= -2 to 0, back upright for x= -4 to -2, upside down again for x= -6 to -4. Do you see that this has period 4?
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Solutions_F11_ECH157_HW3
# Solutions_F11_ECH157_HW3 - d If r → ∞ there will a...
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Unformatted text preview: d) If r → ∞ , there will a large amount of mixing between the two tanks as a result of the very high internal circulation. Thus the process acts like ci q c2 q Total Volume = V1 + V2 Model : dc 2 = q (c i − c 2 ) dt c1 = c2 (complete internal mixing) (V1 +V2) Degrees of freedom analysis is same as part b) Homework Problem Set #3: Solution Key Problem 1: 5.19 a) For the original system, dh1 h = Cqi − 1 dt R1 dh h h A2 2 = 1 − 2 dt R1 R2 A1 where A1 = A2 = π(3)2/4 = 7.07 ft2 ft 3 /min gpm h 2.5 ft = 0.187 3 R1 = R2 = 1 = Cqi 0.1337 × 100 ft /min C = 0.1337 5-23 (9) (10) Using deviation variables and taking Laplace transforms, H 1′ ( s ) = Qi′ ( s ) C = CR1 0.025 = A1 R1 s + 1 1.32 s + 1 1 R1 ′ H 2 ( s) 1 / R1 R2 / R1 1 = = = 1 ′ H 1 (s) A2 R2 s + 1 1.32 s + 1 A2 s + R2 ′ H 2 ( s) 0.025 = Qi′( s ) (1.32s + 1) 2 A1 s + For step change in qi of magnitude M, h1′max = 0.025M ′ h2 max = 0.025M since the second-order transfer function 0.025 is critically damped (ζ=1), not underdamped (1.32 s + 1) 2 2.5 ft Hence Mmax = = 100 gpm 0.025 ft/gpm For the modified system, A dh h = Cq i − dt R A = π(4) 2 / 4 = 12.6 ft 2 V = V1 + V2 = 2 × 7.07ft 2 × 5ft = 70.7ft3 hmax = V/A = 5.62 ft R= h Cq i H ′( s ) = Qi′ ( s ) = 0.5 × 5.62 ft = 0.21 3 0.1337 × 100 ft /min C As + 1 R = CR 0.0281 = ARs + 1 2.64 s + 1 ′ hmax = 0.0281M 2.81 ft Mmax = = 100 gpm 0.0281 ft/gpm 5-24 Hence, both systems can handle the same maximum step disturbance in qi. b) For step change of magnitude M, Qi′( s ) = M s For original system, ′ Q2 ( s ) = 1 1 0.025 M ′ H 2 ( s) = R2 0.187 (1.32 s + 1) 2 s 1 1.32 1.32 = 0.134M − − 2 s (1.32s + 1) (1.32s + 1) t −t / 1.32 q ′ (t ) = 0.134 M 1 − 1 + e 2 1.32 For modified system, Q ′( s ) = 1 1 0.0281 M 2.64 1 H ′( s ) = = 0.134 M − R 0.21 (2.64 s + 1) s s 2.64 s + 1 [ q ′(t ) = 0.134 M 1 − e −t / 2.64 ′ Original system provides better damping since q 2 (t ) < q ′(t ) for t < 3.4. Problem 2: 5.20 a) Caustic balance for the tank, ρV dC = w1c1 + w2 c 2 − wc dt Since V is constant, w = w1 + w2 = 10 lb/min For constant flows, ′ ρVsC ′( s ) = w1C1′ ( s ) + w2 C 2 ( s ) − wC ′( s ) w1 C ′( s ) 5 0.5 = = = ′ C1 ( s ) ρVs + w (70)(7) s + 10 49s + 1 5-25 ′ C m (s) K , = C ′( s ) τs + 1 K = (3-0)/3 = 1 τ ≈ 6 sec = 0.1 min , (from the graph) ′ C m ( s) 1 0.5 0.5 = = C1′ ( s ) (0.1s + 1) (49s + 1) (0.1s + 1)(49s + 1) b) 3 s C1′ ( s ) = 1.5 s (0.1s + 1)(49 s + 1) 1 c ′ (t ) = 1.51 + (0.1e −t / 0.1 − 49e −t / 49 ) m (49 − 0.1) ′ C m ( s) = c) ′ C m ( s) = 0.5 3 1.5 = (49 s + 1) s s (49 s + 1) ( c ′ (t ) = 1.5 1 − e − t / 49 m The responses in b) and c) are nearly the same. Hence the dynamics of the conductivity cell are negligible. 1.5 1 Cm'(t) d) ) 0.5 Part b) Part c) 0 0 20 40 60 80 100 time 120 140 160 Fig S5.20. Step responses for parts b) and c) 5-26 180 200 Two possibilities: 1. K1<0 and K1τ + K2 >0 2. K1 > 0 and K1τ + K2 < 0 e) Gain is negative if K1 < 0 Then zero is RHP if K1τ + K2 > 0 This is the only possibility. f) Constant term and e-t/τ term. g) If input is M/s, the output will contain a t term, that is, it is not bounded. Problem 3: 6.7 a) 2 s −3 −3 2 Q ′( s ) = P ′( s ) = 20 s + 1 20 s + 1 s p ′(t ) = (4 − 2) S (t ) ,P ′( s ) = Q ′(t ) = −6(1 − e −t / 20 ) b) ′ R ′( s ) + Q ′( s ) = Pm ( s ) r ′(t ) + q ′(t ) = p ′ (t ) = p m (t ) − p m (0) m r ′(t ) = p m (t ) − 12 + 6(1 − e − t / 20 ) K= r ′(t = ∞) 18 − 12 + 6(1 − 0) = =6 p (t = ∞) − p (t = 0) 4−2 Overshoot, OS = r ′(t = 15) − r ′(t = ∞) 27 − 12 + 6(1 − e −15 / 20 ) − 12 = = 0.514 r ′(t = ∞) 12 6-7 − πζ OS = exp 1− ζ2 = 0.514 , ζ = 0 .2 Period, T, for r ′(t ) is equal to the period for pm(t) since e-t/20 decreases monotonically. Thus, and c) T = 50 − 15 = 35 τ= T 1 − ζ 2 = 5 .4 6 2π ′ Pm ( s ) K K′ = 22 + P ′( s ) τ s + 2ζτs + 1 τ′s + 1 (K ′τ )s = 2 d) + ( Kτ′ + 2 K ′ζτ) s + ( K + K ′) (τ s 2 + 2ζτs + 1)(τ′s + 1) 2 2 Overall process gain = ′ Pm ( s ) P ′( s ) = K + K′ = 6 −3 = 3 s =0 % psi 6.8 a) Transfer Function for blending tank: Gbt ( s ) = K bt τ bt s + 1 where K bt = τ bt = qin ≠1 ∑ qi 2m 3 = 2 min 1m 3 / min Transfer Function for transfer line Gtl ( s ) = K tl (τ tl s + 1)5 where K tl = 1 τ tl = 6-8 0.1m 3 = 0.02 min 5 × 1m 3 / min t 1 t 2 1 t 3 1 t 4 −t / 6 c5 (t ) = 0.60 − 0.15 1 − e 1 + + + + 6 2! 6 3! 6 4! 6 Using Simulink, b) 0.6 c5 c4 0.58 c3 c2 Concentration 0.56 c1 0.54 0.52 0.5 0.48 0.46 0.44 0 5 10 15 20 25 time 30 35 40 45 50 Figure S6.10. Concentration step responses of the stirred tank. The value of the expression for c5(t) verifies the simulation results above: 52 53 54 c5 (30) = 0.60 − 0.15 1 − e −5 1 + 5 + + + = 0.5161 2! 3! 4! Problem 4: 6.11 a) Y (s) = − τa s + 1 E A B C = + 2+ 2 τ1 s + 1 s ss τ1 s + 1 We only need to calculate the coefficients A and B because Ce − t / τ1 → 0 for t >> τ1. However, there is a repeated pole at zero. 6-13 E (− τ a s + 1) B = lim =E s →0 τ1 s + 1 Now look at E (− τ a s + 1) = As (τ1 s + 1) + B (τ1 s + 1) + Cs 2 − Eτ a s + E = Aτ1 s 2 + As + Bτ1 s + B + Cs 2 Equate coefficients on s: − Eτ a = A + Bτ1 A = − E ( τ a + τ1 ) Then the long-time solution is y (t ) ≈ Et − E (τ a + τ1 ) Plotting (τa+τ1) y y(t)=Et =Et-E(τa+τ1) actual response -E(τa+τ1) time b) For a LHP zero, the apparent lag would be τ1 − τa c) For no zero, the apparent lag would be τ1 6-14 Practice Problems ...
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# What is electric current write its formula and unit?
Contents
Electric current is a measure of the amount of electrical charge transferred per unit time. It represents the flow of electrons through a conductive material. THE SI UNIT OF ELECTRIC CURRENT IS THE AMPERE. AND FORMULA IS 1.V: voltage. 2.I: current.
## What is electric current give its formula?
Electric current is defined the flow of electric charge in a conductor. The SI unit of electric current is Ampere. The formula of electric current is: I = charge/time. (0)
## What is electricity write its unit?
Wikipedia Definition
Electric power is the rate, per unit time, at which electrical energy is transferred by an electric circuit. The SI unit of power is the watt, one joule per second. Electric power is usually produced by electric generators, but can also be supplied by sources such as electric batteries.
## What is current and its unit?
Current is a flow of electrical charge carriers, usually electrons or electron-deficient atoms. … The standard unit is the ampere, symbolized by A. One ampere of current represents one coulomb of electrical charge (6.24 x 1018 charge carriers) moving past a specific point in one second.
## What are types of current?
There are two kinds of current electricity: direct current (DC) and alternating current (AC). With direct current, electrons move in one direction. Batteries produce direct current. In alternating current, electrons flow in both directions.
GOOD TO KNOW: Can you connect two wires with electrical tape?
## What is Ohm’s law state?
Ohm’s law states that the current through a conductor is proportional to the voltage across the conductor. … V=IR where V is the voltage across the conductor and I is the current flowing through it.
## How do I calculate power?
Power is equal to work divided by time. In this example, P = 9000 J /60 s = 150 W .
## What is unit charge?
The unit of charge is the coulomb, and it is defined as the amount of electric charge (q) transported by a constant electric current of one ampere in one second. … In electrostatics, we use the charge of an electron or a proton (Both have the same magnitude of charge) as the unit charge.
## What is the current unit?
Unit of electric current: ampere (A)
The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be 1.602 176 634 ×1019 when expressed in the unit C, which is equal to A s, where the second is defined in terms of ∆νCs.
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# Statistics MCQs –Discrete Distributions Part 4
Get unlimited access to the best preparation resource for UGC : fully solved questions with step-by-step explanation- practice your way to success.
61. Forty percent of the passengers who fly on a certain route do not check in any luggage. The planes on this route seat 20 passengers. What is the probability that 9 or more passengers on a full flight do not check in their luggage?
a. 0.610
b. 0.095
c. 0.944
d. 0.404
e. 0.869
62. Thirty percent of the passengers who fly on a certain route do not check in any luggage. The planes on this route seat 15 passengers. What is the probability that 9 or more passengers on a full flight check in their luggage?
a. 0.610
b. 0.095
c. 0.944
d. 0.404
e. 0.869
63. If X ~ B(6, 0.25), what is P(X > 3)?
a. 0.038
b. 0.004
c. 0.071
d. 0.148
e. 0.031
64. If X ~ B(6, 0.40), what is P(X > 5)?
a. 0.038
b. 0.004
c. 0.071
d. 0.148
e. 0.031
65. If X ~ B(7, 0.25), what is P(X > 3)?
a. 0.038
b. 0.004
c. 0.071
d. 0.148
e. 0.031
66. If X ~ B(7, 0.20), what is P(X > 2)?
a. 0.038
b. 0.004
c. 0.071
d. 0.148
e. 0.031
67. If X ~ B(5, 0.50), what is P(X > 4)?
a. 0.038
b. 0.004
c. 0.071
d. 0.148
e. 0.031
68. Assume that it is known that 80% of monkeys treated with a specific antibiotic recover from a particular disease. If 5 monkeys are treated, find the probability that at least 4 monkeys recover.
a. 0.672
b. 0.328
c. 0.263
d. 0.737
e. 0.583
69. An important part of the customer service responsibilities of a telephone company relates to the speed with which problems in residential service can be repaired. Suppose past data indicate that the probability is 0.5 that problems in residential service can be repaired on the same day. On a given day 5 problems were reported. What is the probability that at least three problems will be repaired on the same day?
a. 0.500
b. 0.031
c. 0.187
d. 0.583
e. 0.261
70. It is known that three out of every ten financial institutions prefer debt-financing to equity-financing. A random sample of twenty financial institutions was selected. What is the probability that at least eight financial institutions prefer debt-financing to equity-financing?
a. 0.7720
b. 0.1130
c. 0.1144
d. 0.2280
e. 0.8870
71. A large manufacturing company that produces CD players believes that 1 out of every 20 CD players is defective. To ensure quality control, a random sample of 120 CD players were selected and tested. A large quality control investigation would be launched if more than 10 out of the 120 CD players selected are defective. What is the expected number of non-defective CD players out of the sample of 120 CD players?
a. 6
b. 114
c. 5
d. 95
e. 120
72. A large manufacturing company that produces CD players believes that 1 out of every 20 CD players is defective. To ensure quality control, a random sample of 120 CD players were selected and tested. A large quality control investigation would be launched if more than 10 out of the 120 CD players selected are defective. What is the expected number of defective CD players out of the sample of 120 CD players?
a. 6
b. 114
c. 5
d. 95
e. 120
73. A large manufacturing company that produces CD players believes that 1 out of every 20 CD players is defective. To ensure quality control, a random sample of 100 CD players were selected and tested. A large quality control investigation would be launched if more than 10 out of the 100 CD players selected are defective. What is the expected number of non-defective CD players out of the sample of 100 CD players?
a. 6
b. 114
c. 5
d. 95
e. 120
74. A large manufacturing company that produces CD players believes that 1 out of every 20 CD players is defective. To ensure quality control, a random sample of 100 CD players were selected and tested. A large quality control investigation would be launched if more than 10 out of the 100 CD players selected are defective. What is the expected number of defective CD players out of the sample of 100 CD players?
a. 6
b. 114
c. 5
d. 95
e. 120
75. At a wholesale protea nursery exactly 100 seeds are planted in each seed-bed, and the probability
that a protea seed will germinate is 0.8. What is the expected number of seeds in the seed-bed that will germinate?
a. 80
b. 96
c. 75
d. 160
e. 65
76. At a wholesale protea nursery exactly 120 seeds are planted in each seed-bed, and the probability
that a protea seed will germinate is 0.8. What is the expected number of seeds in the seed-bed that will germinate?
a. 80
b. 96
c. 75
d. 160
e. 65
77. At a wholesale protea nursery exactly 100 seeds are planted in each seed-bed, and the probability
that a protea seed will germinate is 0.75. What is the expected number of seeds in the seed-bed that will germinate?
a. 80
b. 96
c. 75
d. 160
e. 65
78. At a wholesale protea nursery exactly 200 seeds are planted in each seed-bed, and the probability
that a protea seed will germinate is 0.8. What is the expected number of seeds in the seed-bed that will germinate?
a. 80
b. 96
c. 75
d. 160
e. 65
79. At a wholesale protea nursery exactly 100 seeds are planted in each seed-bed, and the probability
that a protea seed will germinate is 0.65. What is the expected number of seeds in the seed-bed that will germinate?
a. 80
b. 96
c. 75
d. 160
e. 65
80. What is the expected number of heads in 100 tosses of an unbiased coin?
a. 100
b. 25
c. 50
d. 75
e. 0
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## Trigonometry (11th Edition) Clone
$1$
Convert the angle measure to degrees to obtain: $=\frac{\pi}{4} \cdot \frac{180^o}{\pi} = 45^o$ Thus, $\tan{\frac{\pi}{4}} \\= \tan{45^o}$ From Section 2.1 (page 50) , we learned that: $\tan{45^o}=1$
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1. ## Radioactive waste shot into the sun
Proposals for dealing with radioactive waste include shooting it into the Sun. Consider a waster container that is simply dropped from rest in the vicinity of the Earth's orbit. With what speed will it hit the Sun?
Here are some constants that we might need:
G = 6.67 x 10^-11Nm^2/kg^2
Radius (Earth) = 6.38 x 10^6 m
Mass (Earth) = 5.98 x 10^24 kg
Mass = 1.99 x 10^30 kg
1 A.U = 1.49 x 10^11 m
2. Originally Posted by kenan
Proposals for dealing with radioactive waste include shooting it into the Sun. Consider a waster container that is simply dropped from rest in the vicinity of the Earth's orbit. With what speed will it hit the Sun?
Here are some constants that we might need:
G = 6.67 x 10^-11Nm^2/kg^2
Radius (Earth) = 6.38 x 10^6 m
Mass (Earth) = 5.98 x 10^24 kg
Mass = 1.99 x 10^30 kg
1 A.U = 1.49 x 10^11 m
Sounds like a work-energy problem to me.
There is no friction, so there is no non-conservative work going on. Since we have a gravitational potential energy here we need to set a 0 point: I'm going to choose a 0 point out at an infinite distance from the center of the Sun. (Or you could choose one at the center of the Sun, it works out the same. Both choices are commonly used and are equivalent in all respects.)
So:
$W_{nc} = \Delta E$
$\Delta KE + \Delta PE = 0$
$\frac{1}{2}mv^2 - \frac{1}{2}mv_0^2 + -\frac{GmM}{d} - -\frac{GmM}{d_0} = 0$
where m is the mass of the waste, M is the mass of the Sun, d is the radius of the Sun (remember we're measuring all distances from the center of the Sun here), and $d_0$ is the initial distance of the waste from the center of the Sun.
Since the container is "dropped" from rest
$\frac{1}{2}mv^2 - \frac{GmM}{d} + \frac{GmM}{d_0} = 0$
Solve for v.
-Dan
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longest
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https://gpuzzles.com/mind-teasers/classic-five-tablet-puzzle/
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• Views : 80k+
• Sol Viewed : 20k+
# Mind Teasers : Classic Five Tablet Puzzle
Difficulty Popularity
Roy was suffering from severe headaches. He went to see his doctor and the doctor gave him five tablets asking him to take one tablet every 15 minutes.
How much time will it take Roy to consume all the five tablets?
Discussion
Suggestions
• Views : 70k+
• Sol Viewed : 20k+
# Mind Teasers : Most Popular Monty Hall Brain Teaser
Difficulty Popularity
The host of a game show, offers the guest a choice of three doors. Behind one is a expensive car, but behind the other two are goats.
After you have chosen one door, he reveals one of the other two doors behind which is a goat (he wouldn't reveal a car).
Now he gives you the chance to switch to the other unrevealed door or stay at your initial choice. You will then get what is behind that door.
You cannot hear the goats from behind the doors, or in any way know which door has the prize.
Should you stay, or switch, or doesn't it matter ?
• Views : 50k+
• Sol Viewed : 20k+
# Mind Teasers : The Hundred Number Puzzle
Difficulty Popularity
We can create number 100 by using all numbers, i.e.0123456789 and mathematical operators (+/-) in many ways.
Example
98 + 7 + 6 - 5 - 4 - 3 + 2 - 1 = 100
But if we add a condition that the use of the number 21 is the must. Then there are only a few solutions.
One of such solution is:
98 - 7 - 6 - 5 - 4 + 3 + 21 = 100
Can you tell any other solution?
• Views : 70k+
• Sol Viewed : 20k+
# Mind Teasers : Tricky Password Mystery Problem
Difficulty Popularity
A brother wants to use his brother's desktop computer. When he opens it, he finds out that it is password protected. Now, he clicks on the hint and the following appears in front of him:
1 Jug 2 Birthdays 3 Fights 4 Cars 2 Laptops 1 Watch
He is left confused entirely. He has no clue. Can you help him with the password?
• Views : 80k+
• Sol Viewed : 20k+
# Mind Teasers : Palindrome Number Riddle
Difficulty Popularity
There can be myriad ways to create a palindrome. One day, I thought of making my own palindrome. I thought of a number and then decided to add the reversed number to it. Sadly, I did not get a palindrome.
So I kept repeating this step and eventually I succeeded in creating a palindrome. I don't know if you can always create a palindrome using this method but I was able to generate one of four digits.
Can you tell me the number at which I started?
• Views : 90k+
• Sol Viewed : 30k+
# Mind Teasers : IPS Tricky Interview Puzzle
Difficulty Popularity
Jamie looked at his reflection on the window mirror of the 45th floor. Driven by an irrational impulse, he made a leap through the window on the other side. Yet Jamie did not encounter even a single bruise.
How can this be possible if he did neither landed on a soft surface nor used a parachute?
• Views : 50k+
• Sol Viewed : 20k+
# Mind Teasers : Famous Priest Well Puzzle
Difficulty Popularity
An old priest fell on a bottom of the well of a church.
The well is 20 inches deep.
At day time, the old priest climbs 5-inches but when at dark he slip back 4 inches.
In how many days priest will come out of the well ?
• Views : 50k+
• Sol Viewed : 20k+
# Mind Teasers : Tricky Logic Question
Difficulty Popularity
A claustrophobic person boards a train that is just about to enter a tunnel.
Which place will be the best for him to sit?
• Views : 50k+
• Sol Viewed : 20k+
# Mind Teasers : Cricket Riddle
Difficulty Popularity
Two batsman each on 94 runs. Seven runs needed to win in last 3 balls. Both make 100*. How?
• Views : 80k+
• Sol Viewed : 20k+
# Mind Teasers : Police Investigating Murder Riddle
Difficulty Popularity
A couple went to river rafting on a hill station. Two days after their departure, the husband returned alone. He informed the police that her wife was swept away by the waves and died.
On the next day, police arrived to his doorstep and when he opened the door, they arrested him for murdering his wife. They told him that his travel agent had called them.
He was shocked. How did the travel agent know about the murder? Can you suggest how did he know ?
• Views : 70k+
• Sol Viewed : 20k+
# Mind Teasers : Confusing Logical Question
Difficulty Popularity
Sweety have got some Pink and green sock in her drawer.
she had got total of 4 socks.
She pick 2 socks and chances that she get a pair of green socks is 1/2.
What is the chance of her picking a pair of Pink socks ?
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# Stationarity
A common assumption in many time series techniques is that the data are stationary.
A stationary process has the property that the mean, variance and autocorrelation structure do not change over time. Stationarity can be defined in precise mathematical terms, but for our purpose we mean a flat looking series, without trend, constant variance over time, a constant autocorrelation structure over time and no periodic fluctuations (seasonality).
For practical purposes, stationarity can usually be determined from a run sequence plot.
Transformations to Achieve Stationarity
If the time series is not stationary, we can often transform it to stationarity with one of the following techniques.
1. We can difference the data. That is, given the series Zt, we create the new series
Yi=Zi−Z(i−1).
The differenced data will contain one less point than the original data. Although you can difference the data more than once, one difference is usually sufficient.
2. If the data contain a trend, we can fit some type of curve to the data and then model the residuals from that fit. Since the purpose of the fit is to simply remove long term trend, a simple fit, such as a straight line, is typically used.
3. For non-constant variance, taking the logarithm or square root of the series may stabilize the variance. For negative data, you can add a suitable constant to make all the data positive before applying the transformation. This constant can then be subtracted from the model to obtain predicted (i.e., the fitted) values and forecasts for future points.
The above techniques are intended to generate series with constant location and scale. Although seasonality also violates stationarity, this is usually explicitly incorporated into the time series model.
The following plots are from a data set of monthly CO2 concentrations.
The initial run sequence plot of the data indicates a rising trend. A visual inspection of this plot indicates that a simple linear fit should be sufficient to remove this upward trend.
This plot also shows periodical behavior. This is discussed in the next section.
This plot contains the residuals from a linear fit to the original data. After removing the linear trend, the run sequence plot indicates that the data have a constant location and variance, although the pattern of the residuals shows that the data depart from the model in a systematic way.
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https://converths.com/how-many-cm-in-6-inches-online-convertor/
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Let’s walk you through and calculate with video – 6 inches to cm, how many cm is 6 inches?
## How many centimeters in 6 inches?
We usually adopt different units for length measuring in different places. There are several internationally agreed and used systems of measurements. Among them, we have the metric system, Imperial units (also known as British Imperial), and the Chinese system of weights and measures. Each and every system of unit and conversion is common in various countries and regions.
## How long is 6 inches to cm?
But how much cm are 6 inches? As we know, there is 2.54 cm in an inch. Let’s convert 6 inches to centimeters with formulas. 6 inch is equal to how many cm?
So,
. 1 inch = 2.54 centimeters
. or 1 in = 2.54 cm
6 inches = 6in ✖️ 1 in = 6 x 2.54cm = 15.24 centimeters
(PS: cm = centimeter(plural: centimeters), in = inch (plural: inches))
6 inches = 15.24 centimeters
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https://www.studysmarter.us/textbooks/math/precalculus-enhanced-with-graphing-utilities-6th/counting-and-probability/q-53-according-to-the-american-pet-products-manufacturers-as/
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Suggested languages for you:
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Q. 53
Expert-verified
Found in: Page 866
### Precalculus Enhanced with Graphing Utilities
Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465
# According to the American Pet Products Manufacturers Association, there is a 34% probability that a U.S. pet owner owns a cat. If a U.S. pet owner is randomly selected, what is the probability that he or she does not own a cat?
The probability is $0.66$
See the step by step solution
## Step 1. Given Information
The given data is that there is a 34% probability that a U.S. pet owner owns a cat.
## Step 2. Explanation
If E represents an event and $\overline{)E}$represents the complement of E then $P\left(\overline{)E}\right)=1-P\left(E\right)$. It is given that $P\left(E\right)=0.34$
Substitute the given value in the equation and find the required value.
$P\left(\overline{)E}\right)=1-P\left(E\right)\phantom{\rule{0ex}{0ex}}=1-0.34\phantom{\rule{0ex}{0ex}}=0.66$
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https://www.physicsforums.com/search/7631177/
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# Search results
1. ### Determine the value of the inductance ?
Thanks again for the clarification. Hmmmm... why tan(theta) why not cos(theta) ...?? You have answered same question before and i want to understand it 2 .. https://www.physicsforums.com/showthread.php?t=328964 check the link . If i was answering the question i would say the...
yup , i have done it. thanks a lot NvN. all credit goes to u . thanks a ton.
3. ### Determine the value of the inductance ?
wow ..thx a ton rl.bhat. whould u tell what expression must i use if the question says the voltage across the inductor ?? and how did you get Vc-Vl why it not Vl-Vc ?? thanks again ..
4. ### Problem involving the equation of a plane
Here is the problem n = -3i + 3j + 9k it must be -9k Not +9k
5. ### Average force
what does acceleration equal .. ?? Change in what / change in what ... ?? Find the initial and final velocities and change in time and u r done. ^^
6. ### The current in the circuit ?
The current in the circuit ...?? Homework Statement An AC voltage of the form V = Vmax sin 2*pi*f*t , with frequency 22 Hz and maximum voltage 763 V, is applied across a 17 W light bulb. When the voltage is first applied. a) what is the current through the circuit? Three Choices ...
7. ### Determine the value of the inductance ?
Determine the value of the inductance ...?? Homework Statement Consider a series RLC circuit. The applied voltage has a maximum value of 210 V and oscillates at a frequency of 76 Hz. The circuit contains an inductor whose inductance can be varied, a 900 ohm resistor, and a 1 μF...
Ok , i have read the textbook and found the difference b/w the Ip Or J and I I found that the rectangular second moment of area I = pi*(r4)/4 Thanks NvN. Now i have one more question how to find T the torque here : \tauxy = = + T*c / I T = ?? Thanks Pongo38 for the respond i...
9. ### Finding the inductance of a coil
Ideasrule told you that ... B=2.6 mT when I=3.8 A ,,,,,, So find L the length of the solenoid by this equstion : B=μ0*N*I/L Where , B = 2.6e-3 T N = 30 turns I = 3.8 A So L = ? Then substitute the value of L in this equation to find the Inductance of the Solenoid : L...
10. ### Magnetic field between two parallel wires
I think that it is correct for anti-parallel case. 30 A for each wire.
Ok i will not change anything . Hmm , Why it is divided by 4 what is the difference b/w this and J When i calculated J i used R the radius of the shaft not the diameter. Here you divided by 4 which means to me that r here represents the diameter not the radius... am i right ??
http://img268.imageshack.us/img268/1417/1copyr.jpg [Broken] Ok, I switched the z & y axes in my diagram but why u did that ...?? What about the location of stresses are they going to stay on there old localtion after switching...?? (( In this case I think that there woulb no more σy or...
13. ### Simple question about the power generated by an elevator motor
thx for ur responding. So shall i take the nearest answer which is 500 Kg or what ..??
The final Step Find the principal stresses and using the given factor of safety and the yield stress of steel calculate the allowable Load. I think that ,, σx = 0 σy = + M*c / Ip , M = 1.1228e5 *P , c = 0.009525 m , Ip = 1.2929e-8 \tauxy = + T*c / Ip , T = ?? Now...
15. ### Simple question about the power generated by an elevator motor
Homework Statement The motor of an elevator generates 12000 Watts of power when moving the elevator at constant speed of 2 m/s.what is the mass of elevator in kg ...?? There are 5 options for the answer. a) 500 Kg b) 400 Kg c) 300 Kg d) 240 Kg e) 200 Kg Homework...
Now i want to move to step 3. http://img14.imageshack.us/img14/7660/1copyjm.jpg [Broken] Determine the stress state at the wall (location of the maximum bending stress and torsional stress) First of all , Did I locate the stresses correctly with directions , second how to find the...
Ooops i think that I wrote Ig by mistake.Sorry So now Ip is the same found in the 1st step right ..?? Ok the moment now must = 6 * P In SI units = 0.1524 * P So σBending max = 0.1524 * P * 0.009525 / 1.2929e-8 σBending max = 1.1228e5 *P N/m2 Thats it second step done ...
Alright then let's move to the second step . Calculate the maximum bending stress in the circular bar. σBending max = M*c / Ip where M is the bending moment. c is the maximum vertical distance away from the N.A ( Neutral Axis) Ip is the moment of inertia. I found c to be the...
Mechanics Question : Maximum Load needed ?? Urgent Please Help Homework Statement http://img694.imageshack.us/img694/5462/66814773.jpg [Broken] A new steel bracket was designed to support various loads being hung from the end. The designer would like to know the largest load that can be...
20. ### Maximum induced emf with 2 Solenoids
thanks for response hmmm.. ok there is no need to substract one from another, how do i find the flux associated with the outer one ? Do i have to use the inner magnetic field to find the outer flux ?? Or Do i have to use the inner flux to find the outer flux and how ??
21. ### Maximum induced emf with 2 Solenoids
Please anyone who can point me to the right direction ...
22. ### Maximum induced emf with 2 Solenoids
Homework Statement A 3 m long large coil with a radius of 15.7 cm and 130 turns surrounds a 7.2 m long solenoid with a radius of 5.7 cm and 4700 turns. The current in the solenoid changes as I=I_0 sin (2\pi f t) where I0= 30 A and f=60 Hz.Inside solenoid has 4700 turns and outside coil has...
23. ### Magnetic Field , Self-inductance & energy Question
Finally i got the right answer . thanks ideasrule for the help . what i have done to solve this problem : The induced emf for the small solenoid is given by Faraday's Law of induction http://img228.imageshack.us/img228/9731/emf.png [Broken] emf = -N2 dφ/dt = -N2 A2dB/dt =...
24. ### Circuit problem
hi chronicals .. I hope this topic will help you https://www.physicsforums.com/showthread.php?t=248637 Please check it
25. ### Magnetic Field , Self-inductance & energy Question
part d) the emf needed for the small solenoid inside = -n (d[phi]/d[time]) n = 32 turns d[time] = 5 seconds d[phi] = B*A B = μ0 * N * I / L , μ0 is given , N = 323 turns , I = 44 A , L = 0.4 m A = π r^2 , r = 0.002 m after finding d[phi] to be = 5.611e-7 T emf = -...
26. ### Magnetic Field , Self-inductance & energy Question
hmmm ... i answered the first three parts correctly. i missed a little informations in those equations above , that's why i could not answer them. for part a) i used B = μ0 * N * I i forgot the length L so it will become like this B = μ0 * N * I / L for part b ) self inductance...
27. ### Magnetic Field , Self-inductance & energy Question
Homework Statement Given: μ0 = 4 π x10−7 T.m/A. An air-core circular solenoid is shown in the figure below. A current of 22 A is establishedin the wire which makes up this solenoid. http://img412.imageshack.us/i/81565602.jpg/ a) What is the magnetic field at its center? [[ Answer...
28. ### Maximum Induced Emf
thanks kuruman for eplanation. I understood now why i couldn't find a maximum value when deriving the equation. thanks a lot man .
29. ### Maximum Induced Emf
Wow , thanks kuruman for the help , it is right , when time is zero i get a maximum emf E. but i have a question ,,, can i find the maximum by deravatives and how and when to use ? thx in advance for all who are helping us ... ^^
30. ### Capacitor Q=CV Problem
(a) The switch S is open. The top two capacitors are in series: Ctop = 1/(1/3+1/6) = 2 μF, and the bottom two are also in series: Cbot = 1/(1/6 + 1/3) = 2 μF. The top and bottom capacitors are in parallel: Ceq = Ctop + Cbot = 4 μF. The total charge on the capacitors is: Qtot = CeqVab = 840 μC...
31. ### Maximum Induced Emf
Homework Statement A flat loop of wire of area 15.7 cm2 and 1.09 turns is perpendicular to a magnetic field whose magnitude decays in time according to B = 0.5 e−t/7. What is the maximum induced emf? Answer in units of V. Homework Equations \epsilon= \Delta\PhiB/\Deltat \epsilon=...
32. ### Again an electric potential question ?
Wow nice Job Friend , its absolutly right ^^ thank U very much ^^
33. ### Again an electric potential question ?
Aha , i get it now. I will submitt the answer & i am sure it will be correct this time , I'll be right back in minutes ^^ THx again Boss ^^ Really appreciate Ur help !
34. ### Again an electric potential question ?
thx 4 clarifying about the potential ^^ But is'nt taking an integral from 5 to infinity will give me infinity and when i multiply infinity with E i will get also infinity ? In case what u r saying is correct what is the value of r then ?? Is it the distance from point p to x1 or to...
Hi kevinli , welcome to PF ^^ first equation (v=a*t) is for Velocity = acceleration * time in units = (m/s²)*(s) = (m/s) which is the SI unit for Velocity ^^ the second one (d=v*t) is for distance = velocity * time in units = (m/s) * (s) = (m) which is also the SI unit for...
36. ### Again an electric potential question ?
The Attempt at a Solution I know that E in this case = kq/[a*(L+a)] a : distance from point P to x2 ( P-x2) L : distance from x1 to x2 ( x2-x1) The denominator = r²= [a*(L+a)] I want Potential V=kq/r r=Sqr(The denominator = r²= [a*(L+a)]) So I just insert the values and got...
37. ### Again an electric potential question ?
Again an electric potential question ? Homework Statement Question # 1 A charge of 7.034 nC is uniformly distributed along the x-axis from −4 m to 4 m. What is the electric potential (relative to zero at infinity) of the point at 5 m on the x-axis? Homework Equations...
38. ### Block Sliding Down Ramp
I have attached the answers for parts A & B .
39. ### Am i right on this? Newtons secocd law hw problem
Yup , U R right ^^ We are here anytime . Feel Free to ask . May God Help U In your Study ^^
40. ### How To Find Potential ?
Alright Tiny Miny Little GoldFish U got it ^^
41. ### Block Sliding Down Ramp
Wooo Wait friend I faced this problem maybe 2 years ago & actually forgot the whole concept of the problem. So please forgive me, try to read it & understand it by yourself & if U stuck In somewhere tell the Members to clear 4 U that part. Sorry Again Captain.
42. ### How To Find Potential ?
Yup , Finally I got right answer Thx a lot Sir. & thanks also for the clarification about grad ^^ Appreciate it. May God Bless You Sir.
43. ### Block Sliding Down Ramp
Hi captain Evil ^^ I faced this problem before and i kept the answer in my Computer ^^ Download it from here : http://www.mediafire.com/download.php?zw11mdmj1kn [Broken] See Ya
44. ### Am i right on this? Newtons secocd law hw problem
Hi mr.coon ^^ & Welcome to the PF Let positive be west and negative be east since they are opposite directions. The forces are 3300 , -940 , -1200 The sum of the forces : 3300-940-1200= 1160 N Newtons law F=ma is used F = 1160 N m = 6800 Kg a = Find It ^^
45. ### How To Find Potential ?
You said if E=kq/r², then E=-gradU , What do u mean by (gradU) & why it is minus ? and here we have E=kqr/R³, so what V must equal now??
46. ### How To Find Potential ?
Hi again sir ^.* Thx a lot for the information and corection that U've posted & I appreciate your help to me Sir. Part C ) MmmMmmm... I actually know an equation for potential which is V=kq/r but I tried it and got wrong answer. I've tried for r once and for R once & still got wrong...
47. ### How To Find Potential ?
Hi Tiny-Tim! Thx 4 Welcoming ^^ Please check my steps of solving this part & tell me if i made a mistake : \phi=\int\vec{E}.d\vec{A} = Int(EdACos[theta]) = Int(EdA) = EA E : Electric Field inside a uniform sphere of charge = KQr/R3 A : Area of the spherical Shell of...
48. ### How To Find Potential ?
How To Find Potential ?? Homework Statement Consider the uniformly charged sphere with radius R = 4.65461 m, Q = 6.66731 μC is the total charge inside the sphere. a) Find the total flux passing through the Gaussian surface (a spherical shell) with ra- dius 2.08762 m. b) Find...
49. ### Physics problem help please
This Question Is been Repeated but with different values . Please Check THis link it may help you . https://www.physicsforums.com/showthread.php?t=254219
50. ### Can someone me do this problem
Emmm ... Question # 1 Should she try to stop, or should she make a run for it ? We have Her driving speed = 50*(1000/3600) m/s Distance From car to traffic Light = 30 m The time that yellow light can stand = 2.0 s Speed = Distance / Time So Time = Distance / Speed t=d/v...
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# Motion in space velocity and acceleration relationship
### What is acceleration? (article) | Khan Academy
Velocity describes how position changes; acceleration describes how velocity changes. change in speed. But you could also use the steering wheel to turn, which would change your direction of motion. What's the formula for acceleration?. Section Motion in Space: Velocity and Acceleration. Practice HW from To graph the path of the particle, we take the position j i r t t t sin3 cos2)(+.. Position, velocity, and acceleration all describe the motion of an object; Come up with your own twice-differentiable function and draw its graph without a.
As you turn to the left, your acceleration vector points to the left. The dashed line represents the trajectory of an object a car, for example. The acceleration vector points toward the inside of the turn at all times. Now we differentiate this equation: To understand centripetal acceleration, suppose you are traveling in a car on a circular track at a constant speed. Then, as we saw earlier, the acceleration vector points toward the center of the track at all times. This sensation acts in the opposite direction of centripetal acceleration.
The same holds true for non-circular paths. The reason is that your body tends to travel in a straight line and resists the force resulting from acceleration that push it toward the side. This is because the car is decelerating as it goes into the curve.
The tangential and normal components of acceleration can be used to describe the acceleration vector. The tangential and normal unit vectors at any given point on the curve provide a frame of reference at that point. In the following, we ignore the effect of air resistance.
It describes the motion of objects from golf balls to baseballs, and from arrows to cannonballs. First we need to choose a coordinate system. This is the only force acting on the object. An object is falling under the influence of gravity.
How can we modify the previous result to reflect this scenario? First, we can assume it is thrown from the origin. If not, then we can move the origin to the point from where it is thrown.
The horizontal motion is at constant velocity and the vertical motion is at constant acceleration. Motion of a Cannonball During an Independence Day celebration, a cannonball is fired from a cannon on a cliff toward the water.
Find the maximum height of the cannonball. How long will it take for the cannonball to splash into the sea? How far out to sea will the cannonball hit the water?
The flight of a cannonball ignoring air resistance is projectile motion. When the cannonball lands in the water, it is ft below the cannon. Therefore, the cannonball hits the water after approximately The range of the cannon would be determined by finding how far out the cannonball is when its height is ft above the water the same as the altitude of the cannon.
The height of the archer is Find the horizontal distance the arrow travels before it hits the ground. Hint The equation for the position vector needs to account for the height of the archer in meters. In general, what is the maximum distance a projectile can travel, given its initial speed? To determine this distance, we assume the projectile is fired from ground level and we wish it to return to ground level.
In other words, we want to determine an equation for the range. These laws also apply to other objects in the solar system in orbit around the Sun, such as comets e.
### Distance, Velocity, Acceleration
Variations of these laws apply to satellites in orbit around Earth. Kepler's Laws of Planetary Motion The path of any planet about the Sun is elliptical in shape, with the center of the Sun located at one focus of the ellipse the law of ellipses.
The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of the lengths of their semimajor orbital axes the Law of Harmonies. The Sun is located at a focus of the elliptical orbit of any planet. Furthermore, the shaded areas are all equal, assuming that the amount of time measured as the planet moves is the same for each region. We therefore write 1 A. Since the time it takes for Earth to orbit the Sun is 1 year, we use Earth years for units of time.
Then solve for v as a function of t. It's written like a polynomial — a constant term v0 followed by a first order term at. Since the highest order is 1, it's more correct to call it a linear function. It is often thought of as the "first velocity" but this is a rather naive way to describe it. A better definition would be to say that an initial velocity is the velocity that a moving object has when it first becomes important in a problem. Say a meteor was spotted deep in space and the problem was to determine its trajectory, then the initial velocity would likely be the velocity it had when it was first observed.
### Equations of Motion – The Physics Hypertextbook
But if the problem was about this same meteor burning up on reentry, then the initial velocity likely be the velocity it had when it entered Earth's atmosphere. The answer to "What's the initial velocity?
This turns out to be the answer to a lot of questions. The symbol v is the velocity some time t after the initial velocity. Take the case of the meteor. What velocity is represented by the symbol v? If you've been paying attention, then you should have anticipated the answer. It could be the velocity the meteor has as it passes by the moon, as it enters the Earth's atmosphere, or as it strikes the Earth's surface.
## 13.4: Motion in Space: Velocity and Acceleration
It could also be the meteorite's velocity as it sits in the bottom of a crater. Are any of these the final velocity? Someone could extract the meteorite from its hole in the ground and drive away with it.
Probably not, but it depends.
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# AQA Maths FP1 - 15 June 2016 [Exam discussion] watch
1. (Original post by Hjyu1)
I thought that tanx=root 3 as pi(n) +pi/3 are the only possible general solution for it and as tanx had a period of pi you always get root 3. And you matrices question seemed fine as cos/sine have periods of 360 degrees so cos(-60)=cos(300) same for sine
Ahhh actually I think I got root 3!!!
I'm getting mixed up with questions lol
I'm all confused
2. (Original post by Chickenslayer69)
I can't remember exactly :/ But there was an equation with " i + qi " in it and it asked to explain why q must be -1 for it to be real
Basically as the roots were conjugate pairs the cofficent a are all real so the imaginary need to cancel out and only possible real solution for q would be -1
3. (Original post by Hjyu1)
Basically as the roots were conjugate pairs the cofficent a are all real so the imaginary need to cancel out and only possible real solution for q would be -1
Oh, right... I put that imaginary parts had to cancel, would I get a mark for that? Not sure what the other mark is for, stating the roots were conjugate pairs?
4. How many marks was the part b) of the trigonometry question)
And anyone remember what their inequalities were for question 9 last part?
5. Messed up on the summations series question and complex numbers ((((((
6. This is what the graph sketching shouldve looked like, (if the equation i typed in is correct)
7. (Original post by An1998)
Messed up on the summations series question and complex numbers ((((((
Both were a pain in the ass. Lots of people got them wrong so don't worry.
8. (Original post by An1998)
This is what the graph sketching shouldve looked like, (if the equation i typed in is correct)
Omg yes thank god!! I was sure i got that wrong, what did people get for the inequality thing after that? Think I got that wrong lool
9. Q= -12i-1 and 4i-1
P(root 3 over 2, 2)
10. Thought all the questions were easy except all the 6 marker Q's lol
Hardest Fp1 paper personally, all other years papers were a lot easier
11. (Original post by OturuDansay)
Thought all the questions were easy except all the 6 marker Q's lol
Hardest Fp1 paper personally, all other years papers were a lot easier
Agreed, I'm sure a lot of people feel the same People in my class found it hard
12. (Original post by Chickenslayer69)
I think I got -2<k<0.5
What was the equation of C the parabola? I can't remember what it was.
13. (Original post by B_9710)
What was the equation of C the parabola? I can't remember what it was.
This?
14. (Original post by Chickenslayer69)
This?
No the one with the parabola where you had to find possible k values.
15. (Original post by B_9710)
No the one with the parabola where you had to find possible k values.
forgot lol
16. (Original post by B_9710)
What was the equation of C the parabola? I can't remember what it was.
(Y-3)^2=4a(x-2)
17. Here is what I can remember. Someone make an unofficial mark scheme.
I need to focus on M1 & M2 now. I can't dwell on this.
arkhglsigh its going to drive me crazy...
2x^2 + 6x + 3 = 0 ??
alpha + beta= alphabeta=
Roots are alpha/beta & beta/alpha,
Find a quadratic equation with integer coefficients [5 marks]
7x^2 -4x + 7 =0
Linear laws
(a) y=a(b^x)so logy=loga + logb^xlogy=loga + xlogb
You are given that Y= logy and they gave us the graph of Y against x
With the intercept being 2.25??
(b) gradient is -0.4?? I've forgot :/
(i) Values of a & b to 3sf
a is 10^intercept,
General solutions to trigonometric equations
(a) If sin(pi/3) = cos (x/k), find k, k=6 [1 mark]
(b) Find general solution to cos(2x - pi/2) = sin(pi/3) ??
Ended up getting npi + pi/2 and npi + pi/3 ??
(c) Hence find the only finite value for tanx [2 marks]
Can't be pi/2 as tan(90) gives an infinite valueSo pi/3 therefore
tan(60)=root 3
Matrices
Given matrix A(a) Find A^2 [ marks]
(i) What is A^2 geometrically [2 marks]
Stretch in the x-direction by scale factor 4.
(b) Given that the reflection in line x + (root3)y so in the line y=(-1/root 3)x
Find the reflection matrix with everything in its exact trig values. [3 marks]
I found Cos(300) = 1/2 & Sin(300) = -root3/2
(c) Finding coordinates? After A^2 followed by reflection matrix to P(0,-4) [6 marks]
Root 3/2 and 2??
Complex numbers
Quadratic equation z^2 + 4z ... something like that It is given that a root is w= p + 3i ?
For z^2 + (4 + i + iq)z + 20 = 0
(a) Explain why q=-1 [2 marks]
Gets rid of imaginary parts so the coefficient of z is real.
(b) Given that w= p - 3i ?
Find other values for q
Q= -12i-1 and 4i-1
Parabola y^2=4ax, given that it is translated by [2,3] and passes (4,7) ??
(a) Show that a=2 [2 marks] (y-3)^2=4a(x-2), (4)^2=4a(2)16=8a
(b) Show values of k when it does not intersect line ky=x ? [6 marks]
Series question
(a) Show that question [ marks] 3n(4n^2-1) ??
(b) Find four linear factors [6 marks] ??
Sum from r=1 to 2n was in the question.
Graphs of rational functions
Graph: y=(x-1)/(x-2)(2x-1) ??
(a) Find asymptotes [3 marks]y=0, x=2, x=1/2 ??
(b) Sketch the graph [ marks]
y=1/2(x-1) intersects the graph(c) By forming a cubic find values where it intersects?? [ marks]
(d) Hence solve the inequality intersects the graph [3 marks]
18. Same I think although one was a minus on the i
19. (Original post by Pentaquark)
Here is what I can remember. Someone make an unofficial mark scheme.
I need to focus on M1 & M2 now. I can't dwell on this.
arkhglsigh its going to drive me crazy...
2x^2 + 6x + 3 = 0 ??
alpha + beta= alphabeta=
Roots are alpha/beta & beta/alpha,
Find a quadratic equation with integer coefficients [5 marks]
7x^2 -4x + 7 =0
Linear laws
(a) y=a(b^x)so logy=loga + logb^xlogy=loga + xlogb
You are given that Y= logy and they gave us the graph of Y against x
With the intercept being 2.25??
(b) gradient is -0.4?? I've forgot :/
(i) Values of a & b to 3sf
a is 10^intercept,
General solutions to trigonometric equations
(a) If sin(pi/3) = cos (x/k), find k, k=6 [1 mark]
(b) Find general solution to cos(2x - pi/2) = sin(pi/3) ??
Ended up getting npi + pi/2 and npi + pi/3 ??
(c) Hence find the only finite value for tanx [2 marks]
Can't be pi/2 as tan(90) gives an infinite valueSo pi/3 therefore
tan(60)=root 3
Matrices
Given matrix A(a) Find A^2 [ marks]
(i) What is A^2 geometrically [2 marks]
Stretch in the x-direction by scale factor 4.
(b) Given that the reflection in line x + (root3)y so in the line y=(-1/root 3)x
Find the reflection matrix with everything in its exact trig values. [3 marks]
I found Cos(300) = 1/2 & Sin(300) = -root3/2
(c) Finding coordinates? After A^2 followed by reflection matrix to P(0,-4) [6 marks]
Root 3/2 and 2??
Complex numbers
Quadratic equation z^2 + 4z ... something like that It is given that a root is w= p + 3i ?
For z^2 + (4 + i + iq)z + 20 = 0
(a) Explain why q=-1 [2 marks]
Gets rid of imaginary parts so the coefficient of z is real.
(b) Given that w= p - 3i ?
Find other values for qQ= -12i-1 and 4i-1
Parabola y^2=4ax, given that it is translated by [2,3] and passes (4,7) ??
(a) Show that a=2 [2 marks] (y-3)^2=4a(x-2), (4)^2=4a(2)16=8a
(b) Show values of k when it does not intersect line ky=x ? [6 marks]
Series question
(a) Show that question [ marks] 3n(4n^2-1) ??
(b) Find four linear factors [6 marks] ??
Sum from r=1 to 2n was in the question.
Graphs of rational functions
Graph: y=(x-1)/(x-2)(2x-1) ??
(a) Find asymptotes [3 marks]y=0, x=2, x=1/2 ??
(b) Sketch the graph [ marks]
y=1/2(x-1) intersects the graph(c) By forming a cubic find values where it intersects?? [ marks]
(d) Hence solve the inequality intersects the graph [3 marks]
Yes !! I think I got the same as you except I flopped the tan question cause I didn't understand what it was asking at the time but I got the same coordinates for p and the same equation for q.
20. Yes I got exactly that
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Geoboards
This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.
Polydron
This activity investigates how you might make squares and pentominoes from Polydron.
If you had 36 cubes, what different cuboids could you make?
Number Sandwiches
Age 7 to 14 Challenge Level:
Amelie from Burford School, Marlow Bottom in the UK sent in a full explanation:
First, to put one number between 1, two numbers between 2, three numbers between 3, I started by putting 1 at the front. Unfortunately I couldn't find a combination that worked, so I tried putting 2 at the front. It worked...
2 3 1 2 1 3
After that I moved onto the next question: is there more than one way to do it? So I thought I've triend 1 at the front and 2 at the front, so maybe I should put 3 at the front. Yet again it worked...
3 1 2 3 2
As Amelie noticed, this is the same set of numbers backwards.
Elina from Mill Hill County High School in the UK did the opposite but still found both sandwiches:
I began by putting the largest number on the outermost side and positioned the other 2 numbers inside of the sandwich. The formation was:3,1,2,1,3,2. Then I began to see if putting the other 2 numbers (1 and 2) on the outside would also make all 3 fit in if I arranged
them correctly.
Well done also to Shriya from International School Frankfurt, Dheetchanya from Coppell ISD Middle School East in the USA, Ashlynn from ISF Academy in Hong Kong, Rishika from Nonsuch High School for Girls in the UK, Emily T, Emily H, Sam, Lyla and Hattie also from Burford School, Miss Shaw from Cockburn School in the UK, Jadon from Sacred Heart Primary Hemsworth, and Nathan from Bishop Wood C of E Junior School in the UK, who all submitted this correct solution.
Emily T had a slightly different approach:
The first problem required logic. I knew that the threes could fit around the ones' sequence. That only left two twos. I placed a two in between the ones - as the middle number. Then I could only experiment with the remaining card - a two. I tried the two in every possible place and came out with two sequences - 231213 and 312132.
The second problem required more then just logic. I used the aid of the interactivity for this problem. I knew that the ones' and twos' sequence could fit inside the fours. I tried that, but it didn't work as the ones and twos overlapped other numbers. I slid the ones beside the fours, then I put the twos in the possible places - only one worked. The gaps left were three spaces apart - perfect for the threes. I came out with the sequence - 41312432
The third problem was a bit of a mind teaser. Luckily, I had spotted a pattern - all the sequences started with the highest number, followed by a one. I put the seven and the one in these places and then I examined the sequences again . The sequences had a three in between the ones! I slotted the threes in the middle of the ones. I thought about the problem. I realised that the highest numbers would be early in the sequence, as they required more space in between them. The highest number left was six. I placed the sixes in the next possible place. Then I dealt with the fives. I slotted them in next to the sixes, but the last five and last six required the same space. So, I tried the next available space for the fives and it fitted. The next challenge was the fours. I did what I did for the last numbers. I placed the fours in the first available space - luckily it fitted. That left two spaces, two spaces apart . I slotted the only remaining cards in the remaining spaces - the twos. Finally , I had the sequence - 71316435724625.
In conclusion this problem requires patience and logic.
Emily H said:
I began with the number 2, I put both the other numbers in between the number 2’s, therefore I knew that there would be two numbers between them and I could put the other pair of each number either side, so there would be one number between each 2.
Sam noticed that you can't put consecutive numbers next to each other. For example, if you have 2,1,_,_ then the second _ should be a 2 to complete the 2 sandwich but a 1 to complete the 1 sandwich.
For the sandwich with 1, 1, 2, 2, 3, 3, 4, 4, the only sandwich sent in was 4 1 3 1 2 4 3 2 (or the same sandiwch backwards). Well done to Amelie, Dheetchanya, Nathan, Ashlynn, Rishika, Emily T, Emily H, Shriya, Sam, Lyla, Hattie, Elina and Jadon, and to Lyra, Kenny and Olamide, Moyo, Grace, Ryan, Stephen and Nathan, Selah and Jordan from Monarch Global Academy in the USA who all found this sandwich.
Amelie and Elina continued by choosing which number went first. Emily T and Emily H used similar strategies to their strategies for 1, 1, 2, 2, 3, 3. Kenny and Olamide put the largest numbers in first:
To solve, start with the most troublesome number: 4. Figure out the possibilities of where the 4s an go because there are only a couple places they can go. Then fill in the blank spaces between the 4s. Do the 3s next because they’re the second greatest number and there are limits to where they can go. We continued working like that until there were only two spots left for the 1s.
There were lots of possibilites for 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7. Michael S-D from Mill Hill County High School had an approach that nobody had used for 1, 1, 2, 2, 3, 3 or for 1, 1, 2, 2, 3, 3, 4, 4:
I tried to see if it would work starting with a middle ranged number, in this case 5, and then I alternated between small (S), big (B) and middle ranged (M) numbers whilst creating my solution, in the following order: M,S,B,M,S,B,M,M,M,S,B,S,B,M
My 1st solution is: 5 2 7 3 2 6 5 3 4 1 7 1 6 4 or the reverse (4 6 1 7 1 4 3 5 6 2 3 7 2 5)
My 2nd solution is: 3 5 7 4 3 6 2 5 4 2 7 1 6 1 which I found using a similar order to my last solution.
Other correct sandwiches we received were:
5 7 2 3 6 2 5 3 4 7 1 6 1 4 (Ashlynn)
4 6 1 7 1 4 5 2 6 3 2 7 5 3 (Ellees from Burford School)
1 5 1 7 3 4 6 5 3 2 4 7 2 6 (Sam)
4 5 6 7 1 4 1 5 3 6 2 7 3 2 (Hattie)
1 5 1 6 7 2 4 5 2 3 6 4 7 3 (Rishika)
1 7 1 2 6 4 2 5 3 7 4 6 3 5 (Patrick, Charlie and Alex H from Bussindale Primary School in the UK)
5 1 7 1 6 2 5 4 2 3 7 6 4 3 (Shriya and Patrick, Charlie and Alex H, Amelie)
5 7 4 1 6 1 5 4 3 7 2 6 3 2 (Patrick, Charlie and Alex H)
7 2 6 3 2 4 5 3 7 6 4 1 5 1 (Patrick, Charlie and Alex H)
7 3 6 2 5 3 2 4 7 6 5 1 4 1 (Miss Shaw, Nishant from RCHK in Hong Kong)
7 2 4 6 2 3 5 4 7 3 6 1 5 1 (Nathan)
4 6 3 5 7 4 3 2 6 5 2 1 7 1 (Amelie)
5 7 2 6 3 2 5 4 3 7 6 1 4 1 (Amelie)
Emily H, Sam and Shriya tried making number sandwiches with different combinations of numbers. Emily said:
I tried out which numbers I could use to make a complete sandwich:
I could not do it with just 1
I could not with 1 and 2
I could with 1, 2 and 3
I could with 1, 2, 3 and 4
I could not do it with 1, 2, 3, 4 and 5
I could not do it with 1, 2, 3, 4, 5 and 6,
But I could do it with1, 2, 3, 4, 5, 6 and 7!
Shriya gave some examples:
Solution with 1, 2, 4, and 5 but without 3 :
4 5 1 2 1 4 2 5
Solution with 1, 3, 4 and 5 :
3 5 4 1 3 1 4 5
solution with 1, 2, 3, 4, 5, and 7 but without 6 :
7 1 3 1 4 5 3 2 7 4 2 5
Solution with 1, 2, 3 and 6 but without 4 and 5 :
6 3 1 2 1 3 2 6
Solution with 1, 3, 4, 5,and 7 but without 2 and 6 :
4 7 5 3 1 4 1 3 5 7
More solutions are possible when you leave 1 or 2 numbers out.
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# tutorial1 - ERG2020A Tutorial 1 Course Introduction Number...
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ERG2020A Tutorial 1 Course Introduction Number System Lab Introduction
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Course Introduction Lecturer: Prof. Lee Kin Hong (khlee) @ SHB 1017 Tutors: Zhang Yubin, Robin (ybzhang) @ SHB 506 Wang Jinfeng, Phoenix (jfwang) @ SHB 115 Tang Wai Chung, Matthew (wctang) @ SHB 506 Lin Zhenjiang, Allen (zjlin) @ SHB 101
Course Introduction Course webpage http://www.cse.cuhk.edu.hk/~erg2020a Course newsgroup cuhk.cse.erg2020a Please check the webpage and newsgroup daily (or more frequent)!
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Number System With different base, the value would be different even they have the same digits (1100) 2 = (12) 10 (1100) 8 = (576) 10 (1100) 16 = (4352) 10 To avoid confusion, the radix/base must be clearly specified
Number System Positional notation Base b ( a n -1 a n -2 a 0 . a -1 a -2 a - m ) b Polynomial notation (expansion in base 10) a n -1 b n -1 + a n -2 b n -2 + … + a 0 b 0 + … + a - m b - m For example: 29.25 10 = 2×10 1 + 9×10 0 + 2×10 -1 + 5×10 -2 1101.01 2 = 1×2 3 + 1×2 2 + 0×2 1 + 1×2 0 + 0×2 -1 + 1×2 -2
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Base Conversion 10 2, 8, 16 non-10 polynomial notations short div. & mult. bit grouping
Base Conversion Non-10 to 10 Rule: Base- b → polynomial notation → base-10 For example: A3C 16 = A×16 2 + 3×16 1 + C×16 0 (in base-10) = 10×16 2 + 3×16 1
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# preface
Construction problem? Or MO II trial modification? It's okay.
# subject
Main idea of the title:
A group of new guests came to the tavern. In order to entertain them, Bob decided to arrange a wonderful battle!
Bob selected \ (n \) followers \ (a_i \) from \ (m \) followers. Adhering to the principle of simplicity, the combat effectiveness of these \ (m \) followers is an integer between \ ([1,m] \).
In order to make the battle more ornamental, there should not be too few followers participating in the battle, so \ (\ lfloor \ frac{2m}{3} \ rfloor < n \ Le m \).
Bob thinks the best battle is to match two equal opponents! But then he made a mistake, because there were more than two people here. He didn't know how to divide the team! So this task is entrusted to you who works nearby. You need to divide these \ (n \) followers into \ (2 \) groups, and the sum of combat effectiveness of each group is the same.
If there is no solution, it is necessary to report Chaotic evil. Otherwise, NP hard solved is output first, and then \ (- 1 \) and \ (1 \) represent grouping.
$$3\le n\le m \le 10^6.$$ ensure that the sum of the combat effectiveness of Bob's entourage is even.
# explain
Tell a joke, from big to small, violent, greedy dfs has 88pts.
Obviously, there is no idea about the construction problem, so let's talk about the construction method directly.
First, treat \ (n \) as an even number and add \ (0 \) if it is odd.
Then sort and make \ (d_i=a_{2i}-a_{2i-1} \), and you can find \ (\ sum d_i \ Le m - \ frac {n}{2} < n \), and \ (\ sum d_i \) is even.
Let's try to adjust the positive and negative of \ (d_i \) to achieve our goal. We make \ (N=\frac{n}{2} \), and the limit can be expressed as \ (\ sum d_i < 2n \).
Consider inductive construction:
• $$N=1$$, because \ (\ sum d_i < 2n \) and \ (\ sum d_i \) are even.
• $$n > 1$$, if \ (d_i=1 \), there is obviously a solution. Otherwise, we select the largest \ (D {Max} \) and the smallest \ (D {min} \), delete them and add \ (D {Max} - D {min} \). It is not difficult to find that \ (\ sum d_i \) at least reduces \ (2 \), \ (\ sum d_i \) is still even, \ (N'=N-1 \) and is successfully summarized into the sub problem.
The official solution ends here. How can it be realized?
Here are two ideas:
1. We directly simulate this process and consider how to mark the symbol inversion. We can adopt the method of set + heuristic combination. The time complexity \ (O(n\log_2^2n) \) can be achieved if the merging heap is used \ (O(n\log_2n) \), which has not been tried, but the theory can.
2. Despite my rubbish approach, let's look at Mr. Juan's approach. We directly look up the set of \ (2N \) points, \ (i \) and \ (i+N \) represent symbols. Each time we do it, we only need to connect the edges to represent the similarities and differences between the positive and negative signs of numbers. To find the maximum and minimum, we can use set. The time complexity \ (O(n\log_2n) \) and the constant are smaller, and the following code is the same approach.
# code
Small constant
//12252024832524
#include <bits/stdc++.h>
#define TT template<typename T>
using namespace std;
typedef long long LL;
const int MAXN = 1000005;
int n,m,sg;
int a[MAXN],od[MAXN],f[MAXN];
bool ans[MAXN];
{
LL x = 0,f = 1; char c = getchar();
while(c > '9' || c < '0'){if(c == '-') f = -1;c = getchar();}
while(c >= '0' && c <= '9'){x = (x*10) + (c^48);c = getchar();}
return x * f;
}
TT void Put1(T x)
{
if(x > 9) Put1(x/10);
putchar(x%10^48);
}
TT void Put(T x,char c = -1)
{
if(x < 0) putchar('-'),x = -x;
Put1(x); if(c >= 0) putchar(c);
}
TT T Max(T x,T y){return x > y ? x : y;}
TT T Min(T x,T y){return x < y ? x : y;}
TT T Abs(T x){return x < 0 ? -x : x;}
int findSet(int x)
{
if(f[x]^x) f[x] = findSet(f[x]);
return f[x];
}
void unionSet(int u,int v){f[findSet(u)] = findSet(v);}
struct node
{
int val,ID;
bool operator < (const node &px)const{
if(val^px.val) return val < px.val;
return ID < px.ID;
}
};
set<node> s;
int main()
{
// freopen("chaoticevil.in","r",stdin);
// freopen("chaoticevil.out","w",stdout);
for(int i = 1;i <= n;++ i) a[i] = Read();
if(n&1) ++n,sg = 1;
for(int i = 1;i <= n;++ i) f[i] = i,od[i] = i;
sort(od+1,od+n+1,[](int x,int y){
return a[x] < a[y];
});
n >>= 1;
for(int i = 1;i <= n;++ i) s.insert(node{a[od[i<<1]]-a[od[(i<<1)-1]],i});
while(s.size() > 1)
{
auto it1 = s.begin(),it2 = s.end(); --it2;
unionSet(it1->ID,it2->ID+n);
unionSet(it1->ID+n,it2->ID);
node ne = node{it2->val-it1->val,it2->ID};
s.erase(it2); s.erase(s.begin());
s.insert(ne);
}
for(int i = 1;i <= n;++ i)
if(findSet(i)^findSet(1)) ans[od[i<<1]] = 1;
else ans[od[(i<<1)-1]] = 1;
n <<= 1;
printf("NP-Hard solved\n");
for(int i = 1;i <= n - sg;++ i)
{
if(ans[i]) Put(1);
else Put(-1);
putchar(i == (n-sg) ? '\n' : ' ');
}
return 0;
}
Posted on Fri, 05 Nov 2021 17:07:03 -0400 by davey10101
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# 8.1.8 The Ionic Product of Water
### The Ionic Product of Water
#### pH of water
• An equilibrium exists in water where few water molecules dissociate into proton and hydroxide ions
H2O(l) ⇌ H+(aq) + OH(aq)
• The equilibrium constant for this reaction is:
Kc x [H2O] = [H+] [OH]
• Since the concentration the H+ and OH ions is very small, the concentration of water is considered to be a constant, such that the expression can be rewritten as:
Kw = [H+] [OH]
Where Kw (ionic product of water) = Kc x [H2O]
= 10-14 mol2 dm-6 at 298K
• The product of the two ion concentrations is always 10-14 mol2 dm-6
• This makes it straightforward to see the relationship between the two concentrations and the nature of the solution:
[H+] & [OH] Table
#### Worked Example
What is the pH of a solution of potassium hydroxide, KOH(aq) of concentration 1.0 × 10−3 mol dm−3 ?
Kw = 1.0 × 10−14 moldm-6
A. 3
B. 4
C. 10
D. 11
The correct option is D.
• Since Kw = [H+] [OH] , rearranging gives [H+] = Kw ÷ [OH]
• The concentration of [H+] is (1.0 × 10−14) ÷ ( 1.0 × 10−3) = 1.0 × 10−11 mol dm−3
• So the pH = 11
Close
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# PFea Course Lesson 6- Local Region (Breakout) Analysis
Answer to problem from lesson 5
Zooming in closely to any of these nodal loads, they look like a point load applied to a boundary. This has finite load applied at a point, which has zero area, so the applied pressure is infinite. This is like the well-known Boussinesq problem in elasticity. Higher order elements will attempt to solve this singular problem.
In many stress analysis problems the maximum stress in the model is concentrated in a localized region, such as near a fillet, hole, or notch. It is much more economical to “break out” a local region for adaptive analysis rather than reanalyze the entire model. I’ve reviewed the theory behind this previously.
The first step for breakout analysis is determining the origin of the region of interest. This is usually the point of maximum stress in the model but can also be a user-specified point. Starting from that point, the next step is to identify how many elements are needed to retain in the breakout model to get an accurate answer from the adaptive stress analysis. The last step is to apply the displacements from the original solution as constraints at the interface between the break out model and the full model.
The intermediate step of determining how many elements to retain is the trickiest. Ideally, at the interface between the breakout model and the full model, the stress should not be varying too rapidly, and should be much lower than the maximum stress. The elements in the model are sorted in ascending order of distance from their centroids to the origin point. This sorting is very fast if done with the quicksort (“qsort”) algorithm. A minimum number of elements is retained. Originally, additional elements were retained until the stress was low enough compared to the maximum stress or the stress gradient was low enough. Experience showed that both of these tests can often give “false positives” and cause an insufficient number of elements to be retained. Testing also showed that an accurate results is obtained if several thousand elements are retained in the breakout model. So now a very simple approach is used: after the sorting, the first several thousand elements are retained in the breakout model. The default is 5000. A model of this size can be adaptively solved very quickly, typically in a minute or less. This simple algorithm is carried out in postprocess.autoBreakoutSphere.
I refer to this approach of extracting a breakout model from a single large model as “ragged breakout” because the interface between the breakout model and the full model is not smooth. An example of this is shown in the figure below. This turns out not to be problematic, as long as all elements that touch the interface are marked as sacrificial.
Breakout by part from Assemblies
A different approach is used when analyzing assemblies. Here the breakout model is usually a single part. Working with codes that have application-programming interfaces, it is straightforward to determine which part the point of maximum stress is in. Alternatively, the user can select a part to use for the local region. The api can then provide the list of elements that belong to that part. This was done for the interface between SOLIDWORKS Simulation and stressRefine
This information is not available if working with an input file such as the Nastran bulk data file. However, the elements that belong to the part in which the point of maximum stress resides can be determined another way. In Nastran models, the interface conditions between adjacent parts in an assembly are often specified by “glued contacts”. These are “bsurfs” in Nastran, but other codes have a similar interface condition. The boundary of a part consists of element faces that only belong to a single element. These faces are either free (unloaded and unconstrained), loaded, constrained, or at the interface with another part, which means they have “bsurfs” on them. So for the purposes of identifying the part boundary, we just have to look for the faces owned by only one elements, or the faces that have bsurfs. In Nastran models, the faces with bsurfs should only have one element owner also, but checking for the bsurf is still necessary to determine the interface constraints to apply to the model, described below.
The mesh for the part is found by starting at an element in which the point of max stress is found, then searching the mesh topologically: All faces of the starting element are searched. If any are shared with adjacent elements, those elements then have their faces searched. This proceeds until no more faces shared by more than one elements are found. It is easiest to program this recursively, but that might lead to a large number of elements on the stack, so it is done in a loop, with candidate elements added to a list. This is carried out in stressRefine in postprocess. topoFilterSaveBreakoutElems. Here is an example:
Applying interface Conditions
Element faces that are at the interface between the breakout model and the full model have to be identified. The displacements from the FEA solution to the full model are applied as constraints on the nodes of these faces. For a “ragged” breakout model, these are faces that are only owned by one element in the breakout model but are owned by two in the full model. This is carried out in model.FindElemsAdjacentToBreakout. For breakout models that are parts from an assembly, the faces that need the interface condition are those with bsurfs.
Breakout Extraction Architecture
Currently extraction of breakout models is performed by the full stressRefine executable as a special type of run. This executable currently only works on Windows, as discussed previously. It requires the Intel Mkl pardiso solver which is not linking properly to stressRefine on Linux.
For that reason I am going to make a special executable that only does the breakout extraction and does not require an equation solver. This will work on both Linux and Windows. This will have an additional use. Breakout models can be extracted with this executable and then adaptively analyzed with a different program such as Sparselizard by simply providing a translator.
This concludes this PFEA short course. I hope it has been helpful in illustrating some of the issues involved in performing p-adaptive stress analysis, and how they have been handled in stressRefine.
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https://www.storyofmathematics.com/a-uniform-lead-sphere-and-a-uniform-aluminum-sphere-have-the-same-mass/
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# A uniform lead sphere and a uniform aluminum sphere have the same mass. What is the ratio of the radius of the aluminum sphere to the radius of the lead sphere?
The aim of this question is to learn the volume of a sphere and the density of different materials.
If the radius r is known, the volume V of a sphere is given by:
$V \ = \ \dfrac{ 4 }{ 3 } \ \pi r^3 \ … \ … \ … \ (1)$
Also, for a given material the density $d$ is defined as:
$d \ = \ \dfrac{ m }{ V } \ … \ … \ … \ (2)$
Where m is the mass of the body. We will manipulate the above two equations to solve the given problem.
Substituting equation (1) in equation (2):
$d \ = \ \dfrac{ m }{ \bigg ( \ \frac{ 4 }{ 3 } \ \pi r^3 \ \bigg ) }$
$\Rightarrow d \ = \ \dfrac{ 4 m }{ 3 \pi r^3 }$
For lead (say material no. 1 ), the above equation becomes:
$d_1 \ = \ \dfrac{ 4 m_1 }{ 3 \pi r_1^3 } \ … \ … \ … \ (3)$
For Aluminum (say material no. 2 ), the above equation becomes:
$d_2 \ = \ \dfrac{ 4 m_2 }{ 3 \pi r_2^3 } \ … \ … \ … \ (4)$
Dividing and simplifying equation (3) by equation (4):
$\dfrac{ d_1 }{ d_2 } \ = \ \dfrac{ m_1 r_2^3 }{ m_2 r_1^3 }$
Given that:
$m_1 = m_2$
The above equation further reduces to:
$\dfrac{ d_1 }{ d_2 } \ = \ \bigg ( \dfrac{ r_2 }{ r_1 } \bigg )^3 \ … \ … \ … \ (5)$
$\Rightarrow \dfrac{ r_2 }{ r_1 } \ = \ \bigg ( \dfrac{ d_1 }{ d_2 } \bigg )^{ 1/3 }$
From density tables:
$d_1 \ = \ 11.29 \ g/cm^3 \text{ and } d_2 \ = \ 2.7 \ g/cm^3$
Substituting these in equation no. (5):
$\dfrac{ r_2 }{ r_1 } \ = \ \bigg ( \dfrac{ 11.29 }{ 2.7 } \bigg )^{ 1/3 }$
$\dfrac{ r_2 }{ r_1 } \ = \ \bigg ( 4.1814 \bigg )^{ 1/3 }$
$\Rightarrow \dfrac{ r_2 }{ r_1 } \ = \ 1.61$
## Numerical Result
$\dfrac{ r_2 }{ r_1 } \ = \ 1.61$
## Example
Find the ratio of the radiuses of two uniform spheres. One is made up of copper and the other one is made of Zinc.
Let copper and zinc be materials no. 1 and 2, respectively. Then from density tables:
$d_1 \ = \ 8.96 \ g/cm^3 \text{ and } d_2 \ = \ 7.133 \ g/cm^3$
Substituting these in equation no. (5):
$\dfrac{ r_2 }{ r_1 } \ = \ \bigg ( \dfrac{ 8.96 }{ 7.133 } \bigg )^{ 1/3 }$
$\dfrac{ r_2 }{ r_1 } \ = \ \bigg ( 1.256 \bigg )^{ 1/3 }$
$\Rightarrow \dfrac{ r_2 }{ r_1 } \ = \ 1.0789$
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# Circuit with dependent source confusion
Can we find the value of $$\i'\$$ in this circuit, considering $$\I\$$ and $$\R\$$ are known?
Using Kirchhoff’s Current Law we have that $$\I = i + i'\$$.
If the dependent source voltage was $$\ki \;, \; k \neq R \$$ then from Kirchhoff’s Voltage Law we would have $$\(k-R)i=0\$$, and since $$\ k \neq R \$$ we have that $$\i=0\$$, so $$\i'=I\$$.
What happens when $$\ k=R \$$, as in the picture above? Kirchhoff's laws are theoretically satisfied for every value of $$\i\$$. Should we assume that $$\i\$$ is zero again? Or better, does $$\i'\$$ and by extension $$\i\$$ have a fixed value?
If $$\k = R\$$ then the voltage across the dependent source is $$\Ri\$$. The voltage across the resistor is equal to $$\Ri\$$. You are correct that KVL and KCL will be satisfied for all values of $$\i\$$, but that does not mean that you can assume $$\i = 0\$$.
I think you are being asked to think about the value of $$\i'\$$ as a function of $$\i\$$, not as an absolute value. You are almost there.
• So, can $i$ and $i'$ in this circuit take an infinite set of values, with the restriction that they satisfy $I = i+i'$ ? Can't we circuitally conclude, that since the dependent source branch has no resistance, then only this branch draws current from the current source, so $i=0$ ? – ggrin Mar 19 at 17:16
• An ideal voltage source never has resistance, and it doesn't matter. The only thing we know about a voltage source is that it constrains the voltage between its terminals. The voltage source can have a voltage across it even if it has no resistance. Therefore, there can be a voltage across the resistor, and a current through the resistor, and a current through the dependent source. No, you cannot conclude that $i=0$. – Elliot Alderson Mar 20 at 0:41
• Thank you very much!! – ggrin Mar 20 at 21:30
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##### simplify the expression. If not possible,write simplified.
label Algebra
account_circle Unassigned
schedule 1 Day
account_balance_wallet \$5
3x-3 (8-9x)
3 (9m-7)
2-2z+6-5z
3(11a+12f-2f)
Sep 14th, 2014
3x-3(8-9x) = 3x-24+27x = 3x+27x-24 =30x-24 = 3(10x-8)
3(9m-7) = 27m-21
2-2z+6-5z = 2+6-2z+5z= 8+3z
3(11a+12f-2f) = 33a+36f-6f = 33a-30f
Sep 14th, 2014
...
Sep 14th, 2014
...
Sep 14th, 2014
Oct 24th, 2017
check_circle
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# Thread: Evaluate The Following Limit. (1)
1. ## Evaluate The Following Limit. (1)
Hi
Evaluate the following limit :
$\lim_{x\to0} \frac{sinh(x) - sin(x)}{sin^3(x)}$
I used LHospitals, But I stucked.
I tried to use the sandwich theorem:
Clearly $|\frac{sinh(x) - sin(x)}{sin^3(x)}| \leq -(sinh(x)-sin(x))$
Since the minumum value for $sin^3(x)$ is $-1$.
since I make the denominator smaller then the whole fraction is bigger.
and since $-\lim_{x\to0} (sinh(x)-sin(x)) = 0$
then ,By using the sandwich theorem, the desired limit $=0$
Is this right?
Do you have another way to make the limit simpler ?
2. L'hopital's rule will work if you apply it 3 times--there might be a better way
you should get 1/3
3. Originally Posted by Calculus26
L'hopital's rule will work if you apply it 3 times--there might be a better way
you should get 1/3
Thanks.
This means my solution(using sandwich theorem) is wrong.
Still searching for professional solutions.
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Question
# A particle is projected from ground level with an initial velocity of 35m/s at an angle of tan−13/4 to the horizontal. Find the time for which the particle is more than 2m above the ground. [g=10m/s2]
Open in App
Solution
## The angle is given as, tanθ=34 θ=36.86∘ The height is given as, h=(usinθ)t−12gt2 2=21t−5t2 5t2−21t+2=0 The time can be written as, t1+t2=215 t1t2=25 (t1−t2)2=(t1+t2)2−4t1t2 (t1−t2)2=(215)2−4×25 (t1−t2)2=(215)2−4×25 (t1−t2)=4sec Thus, the time for which the particle is more than 2m above the ground is 4sec.
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# THE MOST IMPORTANT PART OF CALCULUS
Everyone needs a little humor (and lots of caffeine) to enjoy Calculus to the fullest. Here are some really cheesy Calculus jokes (some are good) to tickle you pink!
Here's an optimization-ish short story:
One day a farmer called up an engineer and a mathematician and asked them to fence off the largest possible area with the least amount of fence. The engineer made the fence in a circle and proclaimed that he had the most efficient design. The mathematician just laughed at him. She built a tiny fence around herself and said "I declare myself to be on the outside."
I find this one particularly funny:
A guy gets on a bus and starts threatening everybody: "I'll integrate you! I'll differentiate you!!!" So everybody gets scared and runs away. Only one person stays. The guy comes up to him and says: "Aren't you scared, I'll integrate you, I'll differentiate you!!!" And the other guy says: "No, I am not scared, I am ex."
This is good for Calculus students to remember:
Two male mathematicians are in a bar. The first one says to the second that the average person knows very little about basic mathematics. The second one disagrees, and claims that most people can cope with a reasonable amount of math. The first mathematician goes off to the washroom, and in his absence the second calls over the waitress. He tells her that in a few minutes, after his friend has returned, he will call her over and ask her a question. All she has to do is answer one third x cubed. She repeats `one thir -- dex cue'? He repeats `one third x cubed'. Her: `one thir dex cuebd'? Yes, that's right, he says. So she agrees, and goes off mumbling to herself, `one thir dex cuebd...'. The first guy returns and the second proposes a bet to prove his point, that most people do know something about basic math. He says he will ask the blonde waitress an integral, and the first laughingly agrees. The second man calls over the waitress and asks `what is the integral of x squared?'. The waitress says `one third x cubed' and while walking away, turns back and says over her shoulder `plus a constant'!
And that moral of that story is: Don't forget your constants! Those red +C's and -1 on your test/quiz really are not attractive.
Here are the corny ones:
Why do lumberjacks make good musicians?
....because of their natural log-a-rithms!
Mathematics is made of 50 percent formulas, 50 percent proofs and 50 percent imagination.
Calculus students are sometimes clueless. They think that General Calculus was a war hero. If he did actually exist he probably knew how to "integrate" his troops and "differentiate" between his allies and his enemies.
Q: What is the first derivative of a cow?
A: Prime Rib!
This is a good one:
Top ln(e10) reasons why e is better than π:
10) e is easier to spell than π.
9) π = 3.14 while e = 2.718281828459045.
8) The character for e can be found on a keyboard, but π sure can't.
7) Everybody fights for their piece of the pie.
6) ln(pi1) is a really nasty number, but ln(e1) = 1.
5) e is used in calculus while π is used in baby geometry.
4) 'e' is the most commonly picked vowel in Wheel of Fortune.
3) e stands for Euler's Number, π doesn't stand for squat.
2) You don't need to know Greek to be able to use e.
1) You can't confuse e with a food product.
Don't you just hate when this happens...:
"The number you have dialed is imaginary. Please, rotate your phone by 90° and try again..."
AHA! Finally! We can understand the rudiments of mathamatic-ese:
The Dictionary: what mathematics professors say and what they mean by it
Clearly: I don't want to write down all the "in-between" steps.
Trivial: If I have to show you how to do this, you're in the wrong class.
It can easily be shown: No more than four hours are needed to prove it.
Check for yourself: This is the boring part of the proof, so you can do it on your own time.
Hint: The hardest of several possible ways to do a proof.
Brute force: Four special cases, three counting arguments and two long inductions.
Elegant proof: Requires no previous knowledge of the subject matter and is less than ten lines long.
Similarly: At least one line of the proof of this case is the same as before.
Two line proof: I'll leave out everything but the conclusion, you can't question 'em if you can't see 'em.
Briefly: I'm running out of time, so I'll just write and talk faster.
Proceed formally: Manipulate symbols by the rules without any hint of their true meaning.
Proof omitted: Trust me, It's true.
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# Difference between revisions of "Prefix/Infix/Postfix Notation"
The expression $5+\large{8\over{3-1}}$ clearly has a value of 9. It written in infix notation as $5+8/(3-1)$. The value of an infix version is well-defined because there is a well-established order of precedence in mathematics: We first evaluate the parentheses (3-1=2); then, because division has higher precedence that subtraction, we next do 8/2=4. And finally, 5+4=9. The order of precedence is often given the mnemonic of Please excuse my dear Aunt Sue, or PEMDAS: parentheses, exponentiation, multiplication/division, and addition/'subtraction. Multiplication and division have the same level of precedence; addition and subtraction also have the same level of precedence. Terms with equals precedence are evaluated from left-to-right wikipedia.
The algorithm to evaluate an infix expression is complex, as it must address the order of precedence. Two alternative notations have been developed which lend themselves to simple computer algorithms for evaluating expressions. In prefix notation, each operator is placed before its operands5 . The expression above would be 5 8 3 1 - / +. In postfix notation, each operator is placed after its operand. The expression above is + 5 / 8 - 3 1. In prefix and postfix notations, there is no notion of order of precedence, nor are there any parentheses. The evaluation is the same regardless of the operators.
An algorithm for converting from infix to prefix (postfix) is as follows:
• Fully parenthesize the infix expression. It should now consist solely of “terms”: a binary operator sandwiched between two operands.
• Write down the operands in the same order that they appear in the infix expression.
• Look at each term in the infix expression in the order that one would evaluate them, i.e., inner-most parenthesis to outer-most and left to right among terms of the same depth.
• For each term, write down the operand before (after) the operators.
## Example
The following sequence of steps illustrates converting $X=\left(AB-{C\over{D}}\right)^E$ from infix to prefix and postfix:
Infix to Prefix Infix to Postfix
(X = (((A * B) - (C / D)) ↑ E)) (X = (((A * B) - (C / D)) ↑ E))
X A B C D E X A B C D E
X * A B C D E X A B * C D E
X * A B / C D E X A B * C D / E
X - * A B / C D E X A B * C D / - E
X ↑ - *A B / C D E X A B * C D / - E ↑
= X ↑ - * A B / C D E X A B * C D / - E ↑ =
A quick check for determining whether a conversion is correct is to convert the result back into the original format. THhat is, to convert from prefix notation to infox This is best done by changing groups of 2 operands and an operator into a parenthesized infix expression. This needs to be done for us to evaluate the expression easily. Using different examples, here is the process of converting back to infix: Prefix to Infix: Postfix to Infix: ↑ + * 3 4 / 8 2 – 7 5 7 1 + 2 ↑ 7 3 - / 4 + 5 / ↑ + (3 * 4) (8 / 2) (7 – 5) (7 + 1) 2 ↑ (7 – 3) / 4 + 5 / ↑ ((3 * 4) + (8 / 2)) (7 – 5) ((7 + 1) ↑ 2) (7 – 3) / 4 + 5 / (((3 * 4) + (8 / 2)) ↑ (7 – 5)) = 256 ((((7 + 1) ↑ 2) / (7 – 3)) + 4) / 5) = 4
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# Number of smooth numbers less than x
A $$p$$-smooth number is defined as an integer whose prime factors are all less than or equal to $$p$$
In the wiki article about smooth numbers it states:
Let $$\displaystyle \Psi (x,y)$$ denote the number of $$y$$-smooth integers less than or equal to $$x$$ (the de Bruijn function). If the smoothness bound $$B$$ is fixed and small, there is a good estimate for $$\displaystyle \Psi (x,B)$$: $$\displaystyle \Psi (x,B)\sim \frac {1}{\pi (B)!}\prod _{p\leq B}{\frac {\log x}{\log p}}$$ where $$\displaystyle \pi (B)$$ denotes the number of primes less than or equal to $$B$$.
How was this derived? Are there any other good bounds on $$\Psi(x,y)$$?
Let $$S(x, B)$$ be the set of $$B$$ smooth numbers less than or equal than $$x$$. You notice that $$n \in S(x, B)$$ iff $$\log(n) \le \log(x)$$. Writing
$$n = \prod_{p_i \le B} p_i^{\alpha_i}$$
For $$p_i$$ primes and $$\alpha_i$$ the exponent with which appears (possibly zero), $$n\in S(x, B)$$ iff
$$\sum_i \alpha_i \log p_i \le \log(x)$$
This is like asking how many integer coordinates points $$(\alpha_1, \ldots, \alpha_m)$$ are contained in the region
$$A(x, B) = \{ (t_1, \ldots, t_m) \in \mathbb{R}^m : \sum (\log p_i) t_i \le \log(x), \ \ \ t_i \ge 0\}$$
Here $$m =\pi(B)$$ is the number of free parameters we have. It turns out this is closely related to the volume of this region: up to small problems on the boundary, a integer coordinate point contribute with a cube of volume 1, so that
$$\# \{\text{integer coordinate points in } A(x, B) \} \sim \text{volume}(A(x, B))$$
Since the boundary has one dimension less, the approximation gets better and the better as $$x$$ gets larger with respect to $$B$$.
Let's wrap our head around how to calculate this volume. We could do the integral but it's boring. Geometrically, it is a little pyramid of dimension $$m+1$$, with a nice angle at zero and then some edges departing from it. The edges are long $$\log(x) /\log(p_i)$$: you get this by taking the maximum possible $$t_i$$ while all the other parameters are zero, because we are going along an axis.
The volume is linear in the length of each of its edges, so that we can factor out a
$$\prod_{p\le B} \frac{\log(x) }{\log(p) }$$
And we are left with computing the volume of
$$D =\{(t_1, \ldots, t_m) : \sum t_i \le 1, \ \ t_i \ge 0\}$$
For $$m=2$$, this is $$1/2$$. Let's show by induction that the volume of such a pyramid in dimension $$k$$ is $$1/k!$$. If we know this for $$k$$, we also know that a tiny piramid with edges $$=s$$ has volume $$s^k/k!$$, by the same scaling argument as above. For $$k+1$$, we can section along a coordinate and we get pyramids of size $$s$$ for all $$0\le s \le 1$$. Integrating we get
$$\int_{s=0}^1 \frac{s^k}{k! } = \frac{1}{(k+1)! }$$
As desired. Getting back to our original problem, we had $$m=\pi(B)$$ dimensions, so that
$$\Psi(x, B) \sim \frac{1}{\pi(B)! } \prod_{p \le B} \frac{\log x}{\log p}$$
As desired. Note also that you have estimates of both parts in term of $$x, B$$, so that you get a neat estimate in the end!
• Brilliant answer! Would you like to see this question on the number of rough numbers less than or equals to $x$? The Link is below math.stackexchange.com/questions/4002975/… Feb 2, 2021 at 7:40
• Can you elaborate the "right to left" proof of "$n \in S(x,B)$ iff $\log(n) \leq log(x)$" ? Dec 9, 2021 at 1:45
• there's a typo: it should be a "only if". It would be a "iff" if you add "$log(n) \le log(x)$ and all of its prime factors are smaller than $B$" (this condition is actually written in the formulas, because we only consider primes smaller than $B$). Dec 9, 2021 at 14:31
If we let $$y=5$$, for example, each $$5-$$smooth number $$N$$ can be written as $$N=2^a3^b5^c$$. We then have $$\log N=a\log 2 + b\log 3 +c\log 5$$. You can think of a lattice, three dimensional because we have three primes, with each $$5-$$smooth number identified with the point $$(a,b,c)$$ in the lattice. The lattice spacing is $$\log 2, \log 3, \log 5$$ in the three dimensions. $$\Psi(x,5)$$ is then the number of lattice points nearer the origin than a plane that goes through the point $$\log N$$ on each axis. There are $$\frac {\log N} {\log 2}$$ points along the $$2$$ axis and similarly along the other axes. This means the volume of a lattice cell is $$\prod_{p \le 5} \log p$$, which gets us the product in your expression. The fact that we have a tetrahedron instead of a rectangular block give a factor $$\frac 16$$, which corresponds to the $$\frac 1{\pi(B)!}$$ in your expression.
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# Vectors..given that a+b+c=0. Two out of the three vectors are equal in magnitude. The magnitude of the third vector if root2 times that of the either out of the other two. Find the angles b/w these...
Vectors..
given that a+b+c=0. Two out of the three vectors are equal in magnitude. The magnitude of the third vector if root2 times that of the either out of the other two. Find the angles b/w these vectors
quantatanu | Student, Undergraduate | (Level 1) Valedictorian
Posted on
As
a+b+c=0
=> a+b = -c
and so
|a+b| = |c|
that means resultant of vector sum of the two (a & b) is equal to -C.
Now
let
|a|=|b|=a
given
|c|=Sqrt[2] a and is equal to |a+b|
Now resultant of vector sum of a & b is
|a+b| = Sqrt[ |a|^2 + |b|^2 + 2.|a|.|b| Cos(angle between a & b) ]
= Sqrt[ 2 a^2 + 2 a^2 Cos(angle between a & b) ]
=> Sqrt[2] a = Sqrt[2] a Sqrt[1+Cos(angle between a & b) ]
=> Sqrt[1+Cos(angle between a & b) ] = 1
=> 1+Cos(angle between a & b) = 1
=> Cos(angle between a & b) = 0
=> angle between a & b = 90 degrees
and as a & b are equal in magnitude and " - c " is the resultant of vector sum of a & b, so - c lies exactly in the middle of a & b, that is " - c" makes 45 degree angles with a & b, and hence "c" makes "180 - 45" degree angle between a & b,
so
angle between a & b = 90 degrees
angle between a & c = angle between b & c = 180 - 45 degree
= 135 degree
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# Question: What Is Closure Property For Multiplication?
## What is closure property explain with example?
The closure property means that a set is closed for some mathematical operation.
For example, the set of even natural numbers, [2, 4, 6, 8, .
.
.], is closed with respect to addition because the sum of any two of them is another even natural number, which is also a member of the set..
## What is the closure property for polynomials?
CLOSURE: Polynomials will be closed under an operation if the operation produces another polynomial. When adding polynomials, the variables and their exponents do not change. Only their coefficients will possibly change.
## What is the difference between closure property and commutative property?
In summary, the Closure Property simply states that if we add or multiply any two real numbers together, we will get only one unique answer and that answer will also be a real number. The Commutative Property states that for addition or multiplication of real numbers, the order of the numbers does not matter.
## How do you tell if a polynomial set is open or closed?
The test to determine whether a set is open or not is whether you can draw a circle, no matter how small, around any point in the set. The closed set is the complement of the open set. Another definition is that the closed set is the set that contains the boundary or limit points.
## What is closure property formula?
Closure property for addition : If a and b are two whole numbers and their sum is c, i.e. a + b = c, then c is will always a whole number. For any two whole numbers a and b, (a + b) is also a whole number. This is called the Closure-Property of Addition for the set of W.
## What are the 4 properties of math?
The four main number properties are:Commutative Property.Associative Property.Identity Property.Distributive Property.
## Do you add first or multiply first?
Order of operations tells you to perform multiplication and division first, working from left to right, before doing addition and subtraction. Continue to perform multiplication and division from left to right. Next, add and subtract from left to right.
## What are the four basic rules of algebra?
The Basic Laws of Algebra are the associative, commutative and distributive laws. They help explain the relationship between number operations and lend towards simplifying equations or solving them. The arrangement of addends does not affect the sum. The arrangement of factors does not affect the product.
## How can I remember math properties?
Using the name of each property to remember the property itself is the easiest way to keep them straight. Associate the associative property with the word associate. The associative property describes how you can group different sets of numbers together when adding or multiplying with the same result.
## Which operation is not closed for polynomials?
Answer Expert Verified Division. Addition and subtraction are closed for polynomials because the result of adding or multiplying two polynomials is always another polynomial. Division on the other hand is not closed for polynomials; if you divide two polynomials the result is not always a polynomial.
## What is not a polynomial?
A plain number can also be a polynomial term. In particular, for an expression to be a polynomial term, it must contain no square roots of variables, no fractional or negative powers on the variables, and no variables in the denominators of any fractions. … This is NOT a polynomial term…
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# Making Science Graphs and Interpreting Data
Making Science Graphs and Interpreting Data Scientific Graphs Most scientific graphs are made as line graphs. There may be times when other types would be appropriate, but they are rare. The lines on scientific graphs are usually drawn either straight or curved. These "smoothed" lines do not have to touch all the data points, but they should at least get close to most of them. They
are called best-fit lines. In general, scientific graphs are not drawn in connect-thedot fashion. Directly Proportional and Inversely Proportional Graphs Directly Proportional Inversely Proportional As the independent variable increases, the dependent variable increases as well. As the independent variable increases, the dependent variable decreases.
Predicting Data on a Graph Graphs are a useful tool in science. The visual characteristics of a graph make trends in data easy to see. One of the most valuable uses for graphs is to "predict" data that is not measured on the graph. Extrapolate: extending the graph, along the same slope, above or below measured data. Interpolate: predicting data between two measured points on the graph. How to Construct a Line Graph 1. Identify the variables a. Independent variable -Goes on the X axis (horizontal) -Should be on the left side of a data table
b. Dependent variable -Goes on the Y axis (vertical) -Should be on the right side of a data table 2. Determine the scale of the Graph a. Determine a scale (numerical value for each square) that best fits the range of each variable b. Spread the graph to use MOST of the available space How to Construct a Line 3. Graph Number and Label Each Axis a. This tells what the lines on your graph represent. Label each axis with appropriate units.
4. Plot the Data Points a. plot each data value on the graph with a dot. 5. Draw the Graph a. draw a curve or line that best fits the data points. b. Most graphs of experimental data are not drawn as connect the dots. 6. Title the Graph a. Your title should clearly tell what the graph is about. b. If your graph has more than one set of data, provide a key to identify the different lines. Graphing Practice Problem #1a Time (seconds) Distance (meters)
0 0 1 2 2 8 3 18
4 32 5 50 6 72 7 98 8
128 9 162 10 200 A. Graph the data. B. What does the graph represent? Graphing Practice Problem #1b A. What type of motion does this graph
represent? B. Put the data from this graph into a table. Graphing Practice Problem #1c A. Describe what happens during the time represented by this graph. B. Put the data from this graph into a table.
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# Search by Topic
#### Resources tagged with Manipulating algebraic expressions/formulae similar to Shades of Fermat's Last Theorem:
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### How Many Solutions?
##### Stage: 5 Challenge Level:
Find all the solutions to the this equation.
### Reciprocals
##### Stage: 5 Challenge Level:
Prove that the product of the sum of n positive numbers with the sum of their reciprocals is not less than n^2.
### There and Back
##### Stage: 4 Challenge Level:
Brian swims at twice the speed that a river is flowing, downstream from one moored boat to another and back again, taking 12 minutes altogether. How long would it have taken him in still water?
### Always Perfect
##### Stage: 4 Challenge Level:
Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.
### Simplifying Doughnut
##### Stage: 4 and 5 Challenge Level:
An algebra task which depends on members of the group noticing the needs of others and responding.
### Consecutive Squares
##### Stage: 4 Challenge Level:
The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?
### Magic Sums and Products
##### Stage: 3 and 4
How to build your own magic squares.
### Algebra from Geometry
##### Stage: 3 and 4 Challenge Level:
Account of an investigation which starts from the area of an annulus and leads to the formula for the difference of two squares.
### Perfectly Square
##### Stage: 4 Challenge Level:
The sums of the squares of three related numbers is also a perfect square - can you explain why?
### Algebra Match
##### Stage: 3 and 4 Challenge Level:
A task which depends on members of the group noticing the needs of others and responding.
##### Stage: 4 Challenge Level:
Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?
### Three Ways
##### Stage: 5 Challenge Level:
If x + y = -1 find the largest value of xy by coordinate geometry, by calculus and by algebra.
### Never Prime
##### Stage: 4 Challenge Level:
If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.
### ' Tis Whole
##### Stage: 4 and 5 Challenge Level:
Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?
### System Speak
##### Stage: 4 and 5 Challenge Level:
Solve the system of equations: ab = 1 bc = 2 cd = 3 de = 4 ea = 6
### Pair Squares
##### Stage: 5 Challenge Level:
The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.
### DOTS Division
##### Stage: 4 Challenge Level:
Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.
### Telescoping Functions
##### Stage: 5
Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.
### Diverging
##### Stage: 5 Challenge Level:
Show that for natural numbers x and y if x/y > 1 then x/y>(x+1)/(y+1}>1. Hence prove that the product for i=1 to n of [(2i)/(2i-1)] tends to infinity as n tends to infinity.
### Mechanical Integration
##### Stage: 5 Challenge Level:
To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.
### Graphic Biology
##### Stage: 5 Challenge Level:
Several graphs of the sort occurring commonly in biology are given. How many processes can you map to each graph?
### Cosines Rule
##### Stage: 4 Challenge Level:
Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.
### Sums of Pairs
##### Stage: 3 and 4 Challenge Level:
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
### Particularly General
##### Stage: 5 Challenge Level:
By proving these particular identities, prove the existence of general cases.
### Salinon
##### Stage: 4 Challenge Level:
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
### And So on - and on -and On
##### Stage: 5 Challenge Level:
Can you find the value of this function involving algebraic fractions for x=2000?
### Lap Times
##### Stage: 4 Challenge Level:
Two cyclists, practising on a track, pass each other at the starting line and go at constant speeds... Can you find lap times that are such that the cyclists will meet exactly half way round the. . . .
### Complex Partial Fractions
##### Stage: 5 Challenge Level:
To break down an algebraic fraction into partial fractions in which all the denominators are linear and all the numerators are constants you sometimes need complex numbers.
### Really Mr. Bond
##### Stage: 4 Challenge Level:
115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?
##### Stage: 5 Challenge Level:
Find all positive integers a and b for which the two equations: x^2-ax+b = 0 and x^2-bx+a = 0 both have positive integer solutions.
### Sums of Squares
##### Stage: 5 Challenge Level:
Prove that 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, can you prove that twice the sum of two squares always gives the sum of two squares?
### Matchless
##### Stage: 3 and 4 Challenge Level:
There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?
### More Polynomial Equations
##### Stage: 5 Challenge Level:
Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.
### Polynomial Relations
##### Stage: 5 Challenge Level:
Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.
### Look Before You Leap
##### Stage: 5 Challenge Level:
Relate these algebraic expressions to geometrical diagrams.
### Nicely Similar
##### Stage: 4 Challenge Level:
If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?
### Binomial
##### Stage: 5 Challenge Level:
By considering powers of (1+x), show that the sum of the squares of the binomial coefficients from 0 to n is 2nCn
### Fibonacci Factors
##### Stage: 5 Challenge Level:
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
### Janine's Conjecture
##### Stage: 4 Challenge Level:
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
### Poly Fibs
##### Stage: 5 Challenge Level:
A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.
##### Stage: 4 Challenge Level:
If a sum invested gains 10% each year how long before it has doubled its value?
### Root to Poly
##### Stage: 4 Challenge Level:
Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.
### Interpolating Polynomials
##### Stage: 5 Challenge Level:
Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.
### Calculus Countdown
##### Stage: 5 Challenge Level:
Can you hit the target functions using a set of input functions and a little calculus and algebra?
### Unusual Long Division - Square Roots Before Calculators
##### Stage: 4 Challenge Level:
However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?
### Sweeping Satellite
##### Stage: 5 Challenge Level:
Derive an equation which describes satellite dynamics.
### Operating Machines
##### Stage: 5 Challenge Level:
What functions can you make using the function machines RECIPROCAL and PRODUCT and the operator machines DIFF and INT?
### Chocolate 2010
##### Stage: 4 Challenge Level:
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
### The Medieval Octagon
##### Stage: 4 Challenge Level:
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
### Incircles
##### Stage: 5 Challenge Level:
The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?
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median
don steward
mathematics teaching 10 ~ 16
## Saturday, 24 November 2007
### going off at a tangent
why is the tangent to a circle at right angles to the radius?
two methods for establishing this both involve an idea of limit
one way is to establish (RHS) that if you join the middle of a chord to the centre of the circle the line is perpendicular to the chord and then move this chord outwards (parallel to itself) until it just about leaves the circle...
another involves using a chord, extended beyond the circumference at both sides
you can easily show that the two angles that the chord makes with the radiuses are equal (RHS again)
so the supplements (other angle on the straight line) to these angles are equal
again, moving the chord steadily out of the circle shows that these two angles become 90 degrees when the chord becomes a tangent
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https://www.shaalaa.com/question-bank-solutions/which-term-following-sequences-sqrt3-3-3sqrt3-729-geometric-progression-g-p_13671
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| 825,338,824 | 11,066 |
CBSE (Arts) Class 11CBSE
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# Which Term of the Following Sequences: Sqrt3, 3, 3sqrt3, .... is 729 - CBSE (Arts) Class 11 - Mathematics
ConceptGeometric Progression (G. P.)
#### Question
Which term of the following sequences:
sqrt3, 3, 3sqrt3, .... is 729?
#### Solution
Thus, the 12th term of the given sequence is 729.
Is there an error in this question or solution?
#### APPEARS IN
NCERT Solution for Mathematics Textbook for Class 11 (2018 to Current)
Chapter 9: Sequences and Series
Q: 5.2 | Page no. 192
#### Video TutorialsVIEW ALL [1]
Solution Which Term of the Following Sequences: Sqrt3, 3, 3sqrt3, .... is 729 Concept: Geometric Progression (G. P.).
S
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# Give an example of when it would be beneficial to choose one type of discount instead of another.
Posted on
Your question is very vague. I will try to give an example and explain the concept. Assume you go to a supermarket A and you are offered a discount of 10% on a product by the manufacturer and the supermarket offers an additional discount of 5% on the final price. In supermarket B, there is no discount by the manufacturer but the supermarket offers a 15% discount. In both the cases the original price of the product is the same. Should you buy the product from supermarket A or B?
Now if the original price is P, after applying the discounts the final price that you have to pay at supermarket A is P*(1-10%)(1-5%) = P*0.855. In supermarket B, the final price is P*(1-15%) =P*0.85
Therefore though 10 and 5 add up to 15, the price you pay at supermarket B is lesser than what you have to pay at supermarket A.
Posted on
We give two types of discount in the example below:
A mechant gives 10% dicount on TV sets. Another merchant gives a double discount of 6% and over that another 4% . Where to go for our adantage . The second type of double discount looks the same as the 2 discounts the 2nd merchant gives add up to 10%.
So let us calculate if the original price is \$1000.
In the first type we get 10% of 1000 = \$100. we save \$100 over the original price.
In the 2nd type , 6% of \$1000 = \$60.
So the price after 6%discount = \$(1000-60) = \$940.
4% discount over \$940 = \$940*0.04 = \$37.6.
So the final price after discount = 940 - 37.6 = \$902.4.
So the the savings is \$(1000 - 902.4) = \$97.6.
Therefore the first one better from the point of view of purchaser. The second is better for the seller.
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# Existence of Shapes with Irrational Dimensions
Asked by Jon Cypryk, student, A.N. Myer on October 27, 1997:
We would like to pose a question which we have been unable to reach an agreement on. The question is "can a six-sided, 3-D shape exist with a combined surface area of 600 square units? Two opposite sides must have an area of 100 each, two opposite sides with an area of 150 each, and two opposite sides with an area of 50 each."
The tentative answer given is a shape with measurements of a=square root of 75, b=150 divided by the square root of 75, and c=50 divided by the square root of 75. Then a x b = 150 square units, b x c = 100 square units, and a x c = 50 square units.
We understand that this formula works out mathematically because the square root of 75 is rounded off for all intents and purposes because it would be insane not to since it can bring you infinitely close as possible to an area of 600 square units.
However, due to an intense desire to prove or disprove the above statement, we are being precise, exact and very stubborn to the point.
Is it correct that if the length of one of the sides cannot exist (eg., a square of 75), then we could not find an area of one sides equaling exactly and precisely 100 square units? And if this is so then can the shape described in the opening question exist with an exact area of 600 square units?
The item in question would seem to be the length of the a side equaling square of 75.
The two opposing beliefs are:
- the belief that there is no true or exact number for the square root of 75 since it is irrational and continues on forever with no repeating pattern. Therefore side a cannot exist if we choose to be stubborn and exact to the point, which we must in order to get the exact result of 600 without rounding off.
- the belief that even though we cannot get to or see the end of the square root of 75, it it plausible that it can still exist and therefore it is plausible that side a can exist with an exact measurement of square root of 75.
An answer to this would be greatly appreciated if possible to help us prove or disprove the original question.
Jon Cipryk
The real question you are asking is "do irrational numbers exist?" They most certainly do. And so yes, the shape described above does exist.
Several issues need to be addressed to clear up the confusion. First, there are different kinds of numbers. One kind of number is concept of "natural number": the sort of number used to measure "how many". If you were to ask the question "does there exist a number between 1 and 2?", the answer would be "no" if you were referring to the kind of numbers used in counting. For example, it is not possible to press a computer key more than once but less than twice.
However, that does not mean that the number 3/2 does not exist! It just means that it isn't the sort of number used in counting. It exists as a "rational number": a ratio of two integers.
In the same way, a number like the square root of 75 does not exist in the context of rational numbers (just as "half of three" does not exist in the context of the integers). But it does exist in the context of a different number system called the "real numbers" (just as "half of three" does exist in the context of rational numbers).
There are several ways to rigorously define real numbers. One way is to define a real number to be a sequence of rational numbers. So, for example, the sequence of rational numbers 8, 8.6, 8.66, 8.660, 8.6602, ... defines the real number sqrt(75). Note that no individual number in that sequence defines sqrt(75) (which is what you were getting at when you said that no finite decimal exactly equals sqrt(75)); however, the entire sequence taken together defines sqrt(75).
Another way to define real numbers is to define a real number as a partitioning of the rational numbers into two sets, where everything in the first set is less than everything in the second set. (Intuitively, such a partition corresponds to a location on the number line: the place where the first set ends and the second set begins). Now, sqrt(75) corresponds to a perfectly well-defined partition of the rational numbers: for each positive rational number r, either r^2 < 75 or r^2 > 75, and this distinction lets us separate the rationals into two classes, thereby defining a real number if you interpret "real number" as meaning "partition of the rationals into two sets with the appropriate properties".
Each of these definitions is quite abstract. (If you completely understand the previous two paragraphs, you should consider yourself exceptionally gifted in mathematics and I'd encourage you to consider it as a career). Therefore, they are not usually taught until about the third year of an undergraduate program. However, the important thing is that there are such things as "real numbers", that can be rigorously defined (though the definition is abstract and difficult), and within this collection of real numbers there is one whose square is 75. Therefore, the square root of 75 exists.
It's important to realize that these "real numbers" are not just an artificial mathematical construction but are precisely the kind of number system relevant for length measurements (just as the natural numbers are the kind of number system relevant for counting). As a consequence, the square root of 75 exists not just as an abstract mathematical entity, but as a real geometrical length.
One good way to see that some real, physically existing lengths can only be measured by irrational numbers is to think of a square with side length 1. The diagonal of this square (a length that clearly "exists") has length sqrt(2), which is an irrational number.
One final confusion that arises is this: there's a temptation to forget the distinction between a number and a decimal representation of a number. The number 75, for example, is an abstract entity that exists in its own right quite independently of the fact that can be written as the sum 7 x 10 + 5 so that we can write it down as a 7 followed by a 5. When you come across a number like sqrt(75) and observe that it cannot be written down as a finite sum of the above form, meaning that there's no finite decimal representation for it, that doesn't mean the number itself fails to exist. It just means it's a number that happens not to have a finite decimal representation.
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# MS Math – Week of Nov. 14-18, 2022
6th Grade Math – Mrs. VonFeldt and Mrs. Evans
This week in 6th grade math, we will graph ordered pairs on the coordinate system. We have, in past years, graphed in the first quadrant, in which both x and y coordinates are positive (x,y). Now we will graph in 3 other quadrants in which both x and y can be positive or negative. After we practice graphing points, we will find distances between points and solve problems using a coordinate system.
7th Grade Math – Mrs. Evans
In 7th grade math this week we will extend our understanding of proportional relationships by graphing them. We will familiarize ourselves with what proportional relationships look like when they are graphed (straight lines that go through the origin). Proportional relationships are those that have constant rates and can be represented by a unit rate, also known as the constant of proportionality. Here is a video showing how to graph a proportional relationship:
We will finish this module after graphing, review and do an assessment at the end of the week.
8th Grade Math – Mrs. Evans
This week we will finish up Chapter 2 with a project in which we will represent numerous real world situations in equations, tables and graphs. Here is an example of what this looks like:
Upon finishing our projects, we move into Chapter 3, in which we will expand our understanding of exponents, beginning with negative exponents.
Here is a video review of what we should already know about exponents from previous years studies:
Algebra – Mrs. VonFeldt
This week we will wrap up Chapter 3 and test on Thursday. The biggest focus this chapter was on graphing linear equations. We learned how to find the x and y intercepts – two points make a line. And we learned how to graph an equation from slope intercept form: y = mx + b. We also added function notation as a fancy way of identifying functions.
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## Math Olympiad Questions with solutions : Number Systems Class 6
1.The product of 64125 by 173 is_______________ .
(A) 11309625 (B) 11093625 (C) 1130685 (D) 6337505
2.The property satisfied by the division of whole numbers is_______________ .
(A) Closure property (B) Commutative property
(C) Associative property (D) None of these
3.Associative property is used between___________ .
(A) Two numbers (B) Three numbers (C) Four numbers (D) Ten numbers
4.Which of the following set of numbers will make the number sentence true
___________ +___________ = 12.
(A) (6, 8, 12) (B) (6, 8, 16) (C) (16, 8, 10) (D) (6, 8, 10)
5.Which of the following statements is true?
(A) Every whole number is a natural number. (B) Every natural number is a whole number.
(C) 1 is the least whole number. (D) 99 is the largest whole number
6.(38 + 83) + 38 = 38 + (83 + 38) is an example of_________________ property.
(A) Commutative (B) Associative
(C) Closure (D) Distributive
7.If a, b and c are whole numbers such that a > b and c =\ 0 and c > 0, then
(A) a x c < i) x c (B) a x c > b x c
(C) a x c = b x c (D) None of these
8.Difference between the face values of 5 & 9 in 165,234 & 842,928 is___________
(A) 4100 (B) 5900 (C) 4 (D) 14
9.Which of the following properties are NOT applicable to the subtraction of whole numbers?
(A) Closure property (B) Commutative property
(C) Associative property (D) All of these
10.There are 222 red balls in a basket. A boy takes out 6 red balls from it and replaces them by 12 white balls. He continues to do so, till all red balls are replaced by white balls. Then the number of white balls put in the basket is_______
(A) 333 (B) 444 (C) 345 (D) 400
11.Multiplicative identity is______
(A) 1 (B) 0 (C) 2 (D) Both (A) and (B)
12.The number with which 82 is multiplied so that product remains the same is________ .
(A) 82 (B) 0 (C) 1/82 (D) 1
13.If a and b are two whole numbers, then commutative law is applicable to subtraction if and only if .
(A) a = b (B) a =\ b (C) a > b (D) a < b
14.Which of the following operations satisfies the associative law for whole numbers?
(A) Subtraction and division (B) Subtraction and multiplication
(C) Division and multiplication (D) Addition and multiplication
15.If a x b = 0, then _________
A) a =\ 0 (B)b=\0
(C) Either a = 0 or b = 0 (D) Neither a = 0 nor b = 0
16.The expression 6(b + c) is equivalent to 6b + 6c , uses the___________property.
(A) Commutative (B) Closure (C) Distributive (D) Identity
17.Which of the following expressions in INCORRECT?
(A)Positive integer > zero > negative integer
(B)Positive integer > negative integer < zero
(C)Zero < positive integer > negative integer
(D)Positive integer > zero < negative integer
18.Product of two integers is – 48. If one of the integers is – 6, then the other is_____________.
(A) 1 (B) 288 (C) 0 (D) 8
19.The integer which is 2 more than (-7 + (-2)) is_______________ .
(A) -7 (B) -9 (C) 11 (D) -3
20.If positive sign precedes a bracket the sign of the terms inside the bracket will _________ when the bracket is removed.
(A) Not change (B) Change
(C) Be 0 (D) Depends on the bracket
21.Which sign will come in the box to make the expression true?
(-25) – (- 42) – (-27) â–¡ (- 42) – (-25) + (-22)
(A) < (B) > (C) = (D) <
22.Square of any negative integer is____________ .
(A) Negative (B) Positive
(C) 0 (D) Depends on the nature of integer
23.If negative sign precedes a bracket the sign of the terms inside the bracket will when the bracket is removed.
(A) Not change (B) Change
(C) Remains the same (D) Change or remains same
24. Find the sum of the given expression.
(-172) + (-40) + 5 + (-425) + (-275) + 600 -(-15)
(A) 315 (B) -21 (C) 40 (D) -292
25.While removing brackets, the order in which the brackets are removed is____________________ .
(A) [],(),{} (B) {},(),[ ] (C) (),{},[ ] (D) [],{},()
26.Subtract the number obtained by reversing the digits of the number 20198 from the number obtained by interchanging the digits in the unit’s place and the hundred’s place of the same number, we get .
(A) -68211 (B) 68004 (C) -68229 (D) 77121
27.Which of the following is correct ?
(A) -99 < 0 < 2 < -37 (B) -99 < -37 < 0 < 2
(C) -37 < 0 < 2 < -99 (D) -37 < -99 < 0 < 2
28.Multiplying a negative integer for odd number of times gives a_____________ result.
(A) Positive (B) Negative (C) 0 (D) Both (A) and (B)
29.John’s monthly salary is Rs. 12000. He spends Rs. 1450 for his son’s education, Rs. 550 for purchasing clothes, Rs. 450 for purchasing vegetables, milk, etc., Rs. 1500 for purchasing medicine and pays a rent of Rs. 5000 in a particular month. How much does he save in this month?
(A) Rs. 4255 (B) Rs. 4960 (C) Rs.3165 (D) Rs. 3050
30.Which of the following statements is true (if N, W and I are sets of Natural, Whole and Integer numbers respectively ?
(A) N c W c I (B) I c N c W (C) W c N c I (D) I c W c N
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# 352sect17 - Applied Math 352 Autumn Quarter, 2007 R. J....
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Unformatted text preview: Applied Math 352 Autumn Quarter, 2007 R. J. LeVeque 17 Newton’s Method for Nonlinear Systems In Section 12.2 we studied Newton’s method for finding a zero of a function f ( x ) of a single variable x , which is a solution of the equation f ( x ) = 0. This is one equation in one unknown, but generally f ( x ) is a nonlinear function. The idea of Newton’s method is to linearize the equation about the current approximation x ( n ) and solve the resulting linear equation for the next approximation x ( n +1) . Now suppose we have a system of m equations in m unknowns. If the equations are all linear then we can write this in the form Ax- b = ¯ 0 for some matrix A and vector b , and solve the system using the backslash operator. If the equations are nonlinear, then we can use the same idea as for one equation. Now x ( n ) ∈ lR m will be a vector representing an approximation to the solution x * ∈ lR m and we will replace the nonlinear equations by a linearization about x ( n ) that will result in a linear system of m equations. Solving the linear system will give us a new approximation x ( n +1) . A general system of m equations in m unknowns can be written as f ( x ) = ¯ where f is a function mapping lR m to lR m . The i th component of f ( x ) is a function f i ( x ) mapping the vector x to a single real number. Example 17.1. Figure 17.1 shows two circles: one with radius 3 centered at the origin and one of radius 2 centered at ( x 1 , x 2 ) = (1 , 2). Suppose we want to find a point where the circles intersect. Then we need to find a solution ( x * 1 , x * 2 ) to the two equations x 2 1 + x 2 2 = 9 ( x 1- 1) 2 + ( x 2- 2) 2 = 4 . (17.1) This system can be written as f ( x ) = ¯ 0 where f ( x ) = f 1 ( x ) f 2 ( x ) = x 2 1 + x 2 2- 9 ( x 1- 1) 2 + ( x 2- 2) 2- 4 . (17.2) This is a nonlinear function since the components of f cannot be expressed as linear combina- tions of the components of x . For this particular nonlinear system of equations, we can eliminate one of the unknowns fairly easily, reducing the problem to one nonlinear equation in a single unknown that can be solved using Newton’s method. For example, from the first equation of (17.1) we find x 2 = ± q 9- x 2 1 . (17.3) Choosing the + or- sign corresponds to restricting our attention to the upper or lower half...
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# exterior algebras
An exterior algebra is a polynomial ring where multiplication is mildly non-commutative, in that, for every x and y in the ring, y*x = (-1)^(deg(x) deg(y)) x*y, and that for every x of odd degree, x*x = 0.In Macaulay2, deg(x) is the degree of x, or the first degree of x, in case a multi-graded ring is being used. The default degree for each variable is 1, so in this case, y*x = -x*y, if x and y are variables in the ring.
Create an exterior algebra with explicit generators by creating a polynomial ring with the option SkewCommutative.
i1 : R = QQ[x,y,z, SkewCommutative => true] o1 = R o1 : PolynomialRing, 3 skew commutative variable(s) i2 : y*x o2 = -x*y o2 : R i3 : (x+y+z)^2 o3 = 0 o3 : R i4 : basis R o4 = | 1 x xy xyz xz y yz z | 1 8 o4 : Matrix R <-- R i5 : basis(2,R) o5 = | xy xz yz | 1 3 o5 : Matrix R <-- R
i6 : S = QQ[a,b,r,s,t, SkewCommutative=>true, Degrees=>{2,2,1,1,1}]; i7 : r*a == a*r o7 = false i8 : a*a o8 = 0 o8 : S i9 : f = a*r+b*s; f^2 o10 = -2a*b*r*s o10 : S i11 : basis(2,S) o11 = | a b rs rt st | 1 5 o11 : Matrix S <-- S
All modules over exterior algebras are right modules. This means that matrices multiply from the opposite side:
i12 : x*y o12 = x*y o12 : R i13 : matrix{{x}} * matrix{{y}} o13 = | -xy | 1 1 o13 : Matrix R <-- R
You may compute Gröbner bases, syzygies, and form quotient rings of these skew commutative rings.
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0
What is the sum of two numbers that equal 113 and their difference is 13?
Updated: 4/28/2022
Wiki User
12y ago
The two numbers are 50 and 63. Their sum is 113 and they have a difference of 13.
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Q: What is the sum of two numbers that equal 113 and their difference is 13?
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50 and 63
51 and 62
Two numbers whose sum is 138 and whose difference is 88?
138/2 - 44 = 25 138/2 + 44 = 113 113 + 25 = 138 The two numbers are therefore 113 and 25.
25 and 11
What two numbers whose difference is equal to the sum of 538 and 259 with the numbers 17.15.803.25.703.3096?
The sum of 538 and 259 is 797. there are no two numbers among those given with a difference of 797.
What istwo numbers whose difference is equal to the sum of 538 and 259?
There are infinitely many pairs of numbers whose difference is equal to the sum of 538 and 259 (which is 797). One such pair is 420 and 1217. Another pair is 0 and 797.
What is the sum of two numbers between 20 and 40 that is 58 and will equal 12 if subtracted?
The sum of two numbers between 20-40 is 58. The difference is 12
113+7+3 = 123
How do you figure out the sum of to nubers that equal 40 and have a difference of 10?
The two numbers needed are 25 and 15
What 3 prime numbers have the sum of 132?
3 prime numbers which have the sum of 132: 2 + 3 + 127 2 + 17 + 113
The difference of 2 numbers is 32 the sum is 158 what are the 2 numbers?
The difference of 2 numbers is 32 the sum is 158 what are the 2 numbers?
The difference between the two numbers is 8. Their sum is 22. What are the two numbers?
The difference of two numbers is 8. Their sum is 22. What are the two numbers?
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# Verify the Identity 2cot(x)csc(x)=1/(sec(x)-1)+1/(sec(x)+1)
Verify the Identity 2cot(x)csc(x)=1/(sec(x)-1)+1/(sec(x)+1)
Start on the right side.
To write as a fraction with a common denominator, multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Multiply and .
Multiply and .
Reorder the factors of .
Combine the numerators over the common denominator.
Simplify numerator.
Subtract from .
Simplify denominator.
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Apply pythagorean identity.
Convert to sines and cosines.
Apply the reciprocal identity to .
Write in sines and cosines using the quotient identity.
Apply the product rule to .
Simplify.
Multiply the numerator by the reciprocal of the denominator.
Combine and .
Combine.
Cancel the common factor of and .
Rewrite as .
Because the two sides have been shown to be equivalent, the equation is an identity.
is an identity
Do you know how to Verify the Identity 2cot(x)csc(x)=1/(sec(x)-1)+1/(sec(x)+1)? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.
### Name
Name one billion eight hundred seven million eight hundred thirty-seven thousand nine hundred twenty-nine
### Interesting facts
• 1807837929 has 8 divisors, whose sum is 2009392000
• The reverse of 1807837929 is 9297387081
• Previous prime number is 3079
### Basic properties
• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 54
• Digital Root 9
### Name
Name three hundred seventy-eight million eight hundred eighty-six thousand nine hundred eighty
### Interesting facts
• 378886980 has 64 divisors, whose sum is 963532800
• The reverse of 378886980 is 089688873
• Previous prime number is 19
### Basic properties
• Is Prime? no
• Number parity even
• Number length 9
• Sum of Digits 57
• Digital Root 3
### Name
Name one billion six hundred ninety-eight million five hundred eighty-six thousand eight hundred ninety-five
### Interesting facts
• 1698586895 has 4 divisors, whose sum is 2038304280
• The reverse of 1698586895 is 5986858961
• Previous prime number is 5
### Basic properties
• Is Prime? no
• Number parity odd
• Number length 10
• Sum of Digits 65
• Digital Root 2
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Does it even have a true BUILT-IN study guide section? ASVAB PRO contains over 1,000 instructional and tip slides. Other ASVAB apps simply show you questions. What they call studying is nothing more than showing you their questions without grading you. This is the ONLY ASVAB Test Prep app with an actual built-in study guide section covering ALL subjects. With ASVAB PRO you can brush up on topics and learn valuable tips without ever leaving the app.
Para la Fuerza Aérea EE.UU., la intención de volver a probar es que el solicitante para mejorar los últimos resultados del ASVAB por lo que las opciones de alistamiento aumentan. Antes de administrar cualquier nueva prueba, el jefe de vuelo reclutamiento debe entrevistar al solicitante en persona o por teléfono y luego dar su aprobación para la segunda prueba.
#### The Arithmetic Reasoning section of the test measures your ability to solve arithmetic word problems. You may be asked questions such as “If the tire of a car rotates at a constant speed of 552 times in 1 minute, how many times will the tire rotate in half an hour?” Therefore, reviewing common math key words associated with each operation is recommended. For example, if you see the key words “in all,” the problem deals with addition. If the problem asks you to “find the difference,” you are being asked to subtract. If a question asks “how many times” per day or week, you know you are dealing with multiplication. If it asks “how many in each,” you should be thinking about division. The CAT-ASVAB has 16 questions in 39 minutes; the paper-and-pencil version has 30 questions in 36 minutes.
```Understanding the ASVAB score range is to understand standard deviations. The highest score on the ASVAB is a 99 and the lowest score is a one. Scores are based on the mean of all examinees. This is to say that a score of 50 would account for an average score. Each increment of 10 represents a single standard deviation from the mean score. So, for example, a score of 80 would be three standard deviations better than the meanwhile a score of 30 would be two standard deviations lower than the mean score.
```
Cuando se toma la CAT-ASVAB, el ordenador calcula automáticamente e imprime sus puntuaciones estándar para cada subprueba y su línea de resultados para cada tipo de servicio. Esta máquina es una Cookie- muy inteligente también calcula su AFQT en el acto. Con la versión computarizada, que por lo general sabe si califica para el alistamiento militar en el mismo día en que toma la prueba y, si es así, qué puestos de trabajo que puede recibir.
To enlist in the United States armed forces, you must take an entrance examination called the Armed Services Vocational Aptitude Battery (ASVAB). The ASVAB test helps the military determine your qualifications for enlistment. The ASVAB first appeared in 1968. By 1976 it was required by all branches of the military. The test was completely redone in 2002.
The way to prepare for this exam is study hard and then quiz yourself with plenty of practice ASVAB tests. Remember that the exam is identical for all branches, so an Army ASVAB practice test is exactly the same as an ASVAB practice test for the Navy. The most important components of the test are the ones that count towards the Armed Services Qualifications Test, or AFQT. These sections are Word Knowledge, Paragraph Comprehension, Arithmetic Reasoning, and Math Knowledge. For tips and strategies for success on these questions, be sure to review our article on ASVAB Test Prep.
```The questions that have a tendency to arise rather quickly are something along the lines of “why is this test so important?” and “What is the overall purpose of this test?” Well, first it is important to define the actual test and to assess the colorful history of the test. The Armed Services Vocational Aptitude Battery test (ASVAB) is a test that was officially formatted in 1968 with the intention of mentally preparing soldiers with knowledge that identifies with the following:
```
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# Konversi LUASsquare-fathoms ke square-calibers
1 Square Fathoms
= 51852868.217054 Square Calibers
Besaran: luas
Konversi Satuan: Square Fathoms ke Square Calibers
Satuan dasar untuk luas adalah square meters (Non-SI/Derived Unit)
Simbol dari [Square Fathoms] adalah: (sq fath), sedangkan simbol untuk [Square Calibers] adalah: (sq cal), keduanya merupakan satuan dari luas
Perhitungan cepat konversi Square Fathoms ke Square Calibers (sq fath ke sq cal):
1 sq fath = 51852868.217054 sq cal.
1 x 51852868.217054 sq cal = 51852868.217054 Square Calibers.
*catatan: kesalahan atau error kecil dalam pembulatan hasil angka desimal bisa terjadi, silakan dicek ulang.
Definisi:
Berdasarkan satuan/unit dari besaran luas, yaitu => (square meters), 1 Square Fathoms (sq fath) sama dengan 3.34451 square-meters, sedangkan 1 Square Calibers (sq cal) = 6.45E-8 square-meters.
oo
Square Fathomsto Square Calibers (table conversion)
1 sq fath = 51852868.217054 sq cal
2 sq fath = 103705736.43411 sq cal
3 sq fath = 155558604.65116 sq cal
4 sq fath = 207411472.86822 sq cal
5 sq fath = 259264341.08527 sq cal
6 sq fath = 311117209.30233 sq cal
7 sq fath = 362970077.51938 sq cal
8 sq fath = 414822945.73643 sq cal
9 sq fath = 466675813.95349 sq cal
10 sq fath = 518528682.17054 sq cal
20 sq fath = 1037057364.3411 sq cal
30 sq fath = 1555586046.5116 sq cal
40 sq fath = 2074114728.6822 sq cal
50 sq fath = 2592643410.8527 sq cal
60 sq fath = 3111172093.0233 sq cal
70 sq fath = 3629700775.1938 sq cal
80 sq fath = 4148229457.3643 sq cal
90 sq fath = 4666758139.5349 sq cal
100 sq fath = 5185286821.7054 sq cal
200 sq fath = 10370573643.411 sq cal
300 sq fath = 15555860465.116 sq cal
400 sq fath = 20741147286.822 sq cal
500 sq fath = 25926434108.527 sq cal
600 sq fath = 31111720930.233 sq cal
700 sq fath = 36297007751.938 sq cal
800 sq fath = 41482294573.643 sq cal
900 sq fath = 46667581395.349 sq cal
1000 sq fath = 51852868217.054 sq cal
2000 sq fath = 103705736434.11 sq cal
4000 sq fath = 207411472868.22 sq cal
5000 sq fath = 259264341085.27 sq cal
7500 sq fath = 388896511627.91 sq cal
10000 sq fath = 518528682170.54 sq cal
25000 sq fath = 1296321705426.4 sq cal
50000 sq fath = 2592643410852.7 sq cal
100000 sq fath = 5185286821705.4 sq cal
1000000 sq fath = 51852868217054 sq cal
1000000000 sq fath = 5.1852868217054E+16 sq cal
(Square Fathoms) to (Square Calibers) conversions
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https://www.physicsforums.com/threads/p-u-i-why-do-electrons-drop-all-energy-e-u-in-the-circuit.891356/
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# I P=U*I: Why do electrons drop ALL energy e*U in the circuit?
Tags:
1. Oct 30, 2016
### greypilgrim
Hi.
Consider a simple circuit consisting of a voltage source $U$ and a load with resistance $R$, e.g. a lamp or a motor. The current is given by $I=U/R$. The number of electrons passing the circuit per second is $n=I/e$. The power consumed by the load is calculated by
$$P=U\cdot I=U\cdot e\cdot n=\Delta E\cdot n\enspace,$$
where $\Delta E=U\cdot e$ is the (kinetic) energy an electron would gain travelling from the negative to the positive pole of the power source if there was no load.
In this computation, we assume the electron gives all its energy to the load and has kinetic energy zero when it arrives at the plus pole of the power source. But why is that? Why can't the electron maybe only lose half its energy to the load and still have kinetic energy when it enters the battery?
2. Oct 30, 2016
### Staff: Mentor
The kinetic energy of the electrons is pretty much irrelevant for any device other than a particle accelerator.
However, in a bigger picture view of the question, you should not think of energy being stored in individual electrons to be dropped off elsewhere later. The energy is in the fields.
Last edited: Oct 30, 2016
3. Oct 30, 2016
### greypilgrim
But how does the electron know this? Why does it "think" while passing the lamp "I need to give all my energy to the filament such that I arrive at the battery at rest"?
4. Oct 30, 2016
### Staff: Mentor
The electron doesn't know that. Neither does the electron carry energy to give to a load. The fields carry the energy.
5. Oct 30, 2016
### Staff: Mentor
Consider what happens when you turn on a light switch. The change in the fields moves at a little less than the speed of light. By contrast, the electrons move at about a mm/s. How quickly does the energy get from the source to the light? Is it something that happens nearly at the speed of light or closer to a mm/s?
6. Nov 1, 2016
### greypilgrim
Ok, then I think I need a different derivation of $P=U\cdot I$. Because the equation I wrote in #1 starting from the right is basically the derivation I learnt in school, i.e. looking at the energy difference of an electron (or any other current carrier) between the poles of a voltage source and then counting how many of them pass the circuit per second.
How can $P=U\cdot I$ be derived alternatively without looking at individual charges and make assumptions about how much energy they drop in the circuit?
7. Nov 1, 2016
### Staff: Mentor
In terms of circuit theory it should not be derived, it should simply be defined. Then with KVL and KCL you can show that it leads to energy conservation.
The place where you would derive it would be in electromagnetism, after Poynting's theorem is introduced. I like the treatment here, in chapter 11, especially 11.3
http://web.mit.edu/6.013_book/www/book.html
8. Nov 6, 2016
### David Lewis
The current drawn by a motor will be less than V/R unless the motor shaft is prevented from turning. Also, the voltage across the load should be used, not the voltage of the source.
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# Non-Exact Equation - Integrating Factor
• Oct 10th 2010, 02:08 PM
Kasper
Non-Exact Equation - Integrating Factor
Hey I've got a question here where I need to find an integrating factor to make an equation exact, but it's in two variables and I'm having trouble finding both values!
Quote:
For what values of m and n will $\displaystyle u=x^ny^m$ be an integrating factor for the differential equation
$\displaystyle (-12y+14x)dx + (4x-6x^2y^{-1})dy=0$
To do this i'm using the exactness formula $\displaystyle \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ Which gives me:
$\displaystyle -12(m+1)x^ny^m+14mx^{n+1}y^{m-1}=4(n+1)x^ny^m - 6(n+2)x^{n+1}y^{m-1}$
$\displaystyle -12(m+1)+14m=4(n+1)-6(n+2)$
$\displaystyle m=2-n$
But i'm not sure where to go from here to get another equation to find m and n. A push in the right direction would be awesome. Thanks for your help!
-Kasper
• Oct 10th 2010, 03:35 PM
Krizalid
• Oct 10th 2010, 06:03 PM
Kasper
Beauty thanks Kryzalid!
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20.4 C
New York
Friday, September 24, 2021
# Designing a Logic System
In Module 1 we saw the principles of simple logic design. We saw how we could produce a truth table from a combination of different logic gates. The diagram shows another combination of logic gates:
Question 1. Complete the truth table and click to see the answer
A B C D
0 0 1 0
0 1 1 1
1 0 0 0
1 1 0 0
In drawing up the truth table, we notice that:
• C is A inverted, so that when A is a 1, C is a 0 and vice versa.
• D is a 1 when both C AND D are a 1.
• Q is a 1 when B is a 1.
Now let us do the reverse. We are going to design a circuit having been given a description.
Design a circuit with two inputs A and B that will give an output that is high only when B is high and A is low.
A B Q 0 0 0 0 1 1 1 0 0 1 1 0
1. We know that an AND gate will give a 1 only when both its inputs are high. So we can connect B directly to the input, but not A. We have a clue in the last sentence, NOT A.
1. So we need a NOT gate in the line from A.
Question 2. Can you draw this circuit?
We can solve a problem if we are given a truth table rather than a written description. Here is another truth table:
A B Q 0 0 1 0 1 0 1 0 1 1 1 1
Question 3. What three conditions are needed for the outputs to be 1?
Question 4. Can you draw this circuit? C
## Simplifying Logic Circuits with Boolean Algebra
The methods we have looked at above are OK for very simple systems, but with a more complex system we would find the truth tables very cumbersome, if not impossible. So we can use Boolean algebra, which we first met in Module 1. Let us look at a few of the rules:
• AB, and Q represent the variables which can only be 1 or 0.
• (“A-bar”) is NOT A. If AŻ = 0, A = 1 and vice versa. is said to be complementary to A.
Important Note: It is impossible using this web editor to place the bar above the A, so I have placed it behind, like this, . In Word, I have put the bars above the letters using the line drawing function. However it does not copy across at all well. In some questions I have written “A-bar”. If you can do it any better, then please tell me, and then come and do the edits for me! The double compliment A bar-bar I will write as AŻ Ż.
Go back to Module 1 Topic 2 to revise the basic rules of Boolean Algebra
A useful property of logic gates is that there is often more than one solution to a problem, and Boolean algebra helps to simplify the expressions required. This allows us to use the minimum number of gates, desirable as that reduces the effort in design and wiring. Here are some laws that will simplify matters:
• De Morgan’s First Law:
• De Morgan’s Second Law
There are other laws:
• Double inversion:
• Commutative Laws:
A + B = B + A
• Associative Laws:
A + (B + C) = (A + B) + C
A.(B.C) = (A.B).C
• Distributive Laws:
A .(B + C) = (A.B) + (A.C)
A + (B.C) = (A + B).(A + C)
• Product of Sums:
(A + B).(A + B) = A.A + A.B + B.A + B.B
• Redundancy:
A.B + A.B.C + A.B.D = A.B
There are a lot of laws here, which we will use to simplify the circuit as shown in below:
The Boolean expression for this circuit is:
The associative law says:
See where the brackets have moved. Well so what? Look at De Morgan I:
Then De Morgan II gives us:
The double bar gives us a double inversion, so we get:
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Adult Numeracy, Functional Maths, and GCSE Resources
Displaying 1 - 10 of 503 resources:
Properties of 2D Shapes - quadrilaterals and triangles
Name each 2D shape and describe its properties. Two separate worksheets: one for triangles, one for quadrilaterals. For Entry Level, hence angles are not a focus.
For each of the 6 quadrilaterals and 4 triangles, the learner is prompted to answer these questions:
Level
E2
E3
Maths
FM Context free underpinning
E2.19 Recognise and name 2-D and 3-D shapes including pentagons, hexagons, cylinders, cuboids, pyramids, spheres
E2.20 Describe properties of common 2-D & 3-D shapes including nos. of sides, corners, edges, faces, angles & base
E3.19 Sort 2-D and 3-D shapes using properties including lines of symmetry, length, right angles, angles including in rectangles and triangles
Properties of 3D Objects for Entry Level Functional Maths
Name each 3D object and fill in the gaps about their properties.
For each of the 9 common 3D shapes, the learner is prompted to complete information about:
• its name
• the number of curved and/or flat faces
• the shape of the faces
• the number of vertices (corners)
• the number of edges
Editor's note
Answer sheet with detailed Functional Skills Maths mapping is available to contributors only
Level
E1
E2
E3
Maths
FM Context free underpinning
E1.9 Identify & recognise common 2-D and 3-D shapes including circle, cube, rectangle (incl. square) and triangle
E2.19 Recognise and name 2-D and 3-D shapes including pentagons, hexagons, cylinders, cuboids, pyramids, spheres
E2.20 Describe properties of common 2-D & 3-D shapes including nos. of sides, corners, edges, faces, angles & base
E3.19 Sort 2-D and 3-D shapes using properties including lines of symmetry, length, right angles, angles including in rectangles and triangles
Properties of 2D Shapes for Entry Level Functional Maths
Name each 2D shape and describe its properties.
For each of the 8 common shapes, the learner is prompted to answer these questions:
Level
E1
E2
E3
Maths
FM Context free underpinning
E1.9 Identify & recognise common 2-D and 3-D shapes including circle, cube, rectangle (incl. square) and triangle
E2.19 Recognise and name 2-D and 3-D shapes including pentagons, hexagons, cylinders, cuboids, pyramids, spheres
E2.20 Describe properties of common 2-D & 3-D shapes including nos. of sides, corners, edges, faces, angles & base
E3.19 Sort 2-D and 3-D shapes using properties including lines of symmetry, length, right angles, angles including in rectangles and triangles
L2 Currency Conversions
This is a Level 2 based Currency Conversion set of 4 questions with resource sheets and answers. All exchange rates resources were obtained from web sites such as The Post Office and the Bank of England.
Editor's note
Also provides useful practice in multiply, dividing with decimals (up to 4 decimal places); sensible rounding of answers; and extracting data from tables.
Level
L1
L2
GCSE L1-5
Maths
FM Straightforward problem(s) with more than 1 step
FM Complex multi-step problem(s)
L2.13 Calculate amounts of money, compound interest, percentage increases, decreases and discounts including tax and simple budgeting
FM L2.14 Convert between metric and imperial units of length, weight and capacity using a) a conversion factor and b) a conversion graph
L2.10 Add, subtract, multiply and divide decimals up to three decimal places
Context
Leisure Travel Tourism
Area and Perimeter Level 1
Functional Skills level 1. Revision booklet for area and perimeter. Covers rectangles and compound shapes made up of rectangles.
Editor's note
A very useful set of questions with guidance. Lots of underpinning practice is followed by problem solving. Ideal for non-calculator work. Note there is no answer sheet.
Level
L1
Maths
L1.22 Calculate area and perimeter of simple shapes including those that are made up of a combination of rectangles
FM Context free underpinning
FM Straightforward problem(s) with more than 1 step
Leap year Functional Maths challenge
A hastily written resource (I didn’t want to have to wait another 4 years!) on Feb 29th 2020 (but can be used throughout the leap year). Covers Functional Skills (Measures) content descriptors relating to using dates and units of time. Number topics such as estimation and checking, multiplication, division, odd and even numbers, and sequences are also included. There is an emphasis is on using non-calculator written methods to convert between units of time.
Level
E2
E3
L1
L2
Maths
FM Contextualised underpinning
FM Simple one step problem(s)
FM Straightforward problem(s) with more than 1 step
FM L1.4 Use multiplication facts and make connections with division facts
L2.2 Carry out calculations with numbers up to one million including strategies to check answers including estimation and approximation
E2.13 Read and record time in common date formats, and read time displayed on analogue clocks in hours, half hours and quarter hours, and understand hours from a 24-hour digital clock
FM L1.20 Convert between units of length, weight, capacity, money and time, in the same system
General
Generic resources for literacy, numeracy and beyond
Pancake Proportions
This resource covers all the Functional Skills content descriptors relating to ratio and proportion. It was written with mixed L1-L2 classes in mind. I wanted to experiment with inverse proportion questions (a new topic in Reformed Functional Maths) but wanted to build up to them gradually. The questions are contextualised and problem based, with no underpinning taught, so learners will need an introduction to ratios (or a refresher) beforehand. Ratios, direct and inverse proportion are covered.
Level
L1
L2
GCSE L1-5
Maths
FM Straightforward problem(s) with more than 1 step
FM Complex multi-step problem(s)
FM L1.17 Work with simple ratio and direct proportions
FM L2.11 Understand and calculate using ratios, direct proportion and inverse proportion
Context
Catering Food Nutrition
HS2 Compass points, position and bearings
Covers all reformed Functional Skills content descriptors relating to position and direction (from E1 to L1). Written with mixed-level classes in mind. Positional vocabulary; cardinal and intercardinal compass points; turns (e.g. quarter turns) and bearings are all covered in this set of graduated HS2 (high speed railway) themed problems. Problems are interspersed with context-free underpinning questions, examples and tips.
Level
E1
E2
E3
L1
Maths
FM Context free underpinning
FM Simple one step problem(s)
FM Straightforward problem(s) with more than 1 step
E1.10 Use everyday positional vocabulary to describe position and direction including left, right, in front, behind, under and above
E2.21 Use appropriate positional vocabulary to describe position and direction including between, inside, outside, middle, below, on top, forwards and backwards
E3.20 Use appropriate positional vocabulary to describe position and direction including eight compass points and including full/half/quarter turns
L1.26 Use angles when describing position and direction, and measure angles in degrees
Context
Leisure Travel Tourism
Motor vehicles & Transport
HS2 Journey times
A topical resource that requires students to interpret a news infographic and calculate the difference between two times. There is also a second question for Level 2 learners on Speed, Distance, Time. As well as enabling students to practise time calculations it can lead to discussion about this controversial rail project.
Editor's notes
Really topical and interesting. With curriculum mapping. No answer sheet.
Level
L1
L2
Maths
FM Straightforward problem(s) with more than 1 step
FM Complex multi-step problem(s)
FM L1.20 Convert between units of length, weight, capacity, money and time, in the same system
L2.15 Calculate using compound measures including speed, density and rates of pay
Context
Leisure Travel Tourism
Motor vehicles & Transport
Vending Machine Functional Maths
A set of Entry Level tasks - all based on a vending machine. The main focus is money but positional vocabulary is also covered. Skills covered include: recognising letters and numbers for items in a vending machine, identifying coins to pay with, working out change from £1, checking answers, rounding and estimating, distinguishing between right and left.
Editor's note
A delightfully functional resource. With teaching notes. Fully mapped to the Reformed Functional Skills content.
Level
E1
E2
E3
Maths
FM Simple one step problem(s)
FM E1.1 Read, write, order and compare numbers up to 20
E2.2 Read, write, order and compare numbers up to 200
FM E2.5 Add and subtract two-digit numbers
E1.5 Recognise coins and notes and write them in numbers with the correct symbols (£ & p), where these involve numbers up to 20
E2.12 Calculate money with pence up to one pound and in whole pounds of multiple items and write with the correct symbols (£ or p)
FM E3.10 Calculate with money using decimal notation & express money correctly in writing in pounds and pence
FM E3.11 Round amounts of money to the nearest £1 or 10p
E1.10 Use everyday positional vocabulary to describe position and direction including left, right, in front, behind, under and above
E2.21 Use appropriate positional vocabulary to describe position and direction including between, inside, outside, middle, below, on top, forwards and backwards
Context
Independent living
Catering Food Nutrition
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# GMAT prep quant! i need help
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22 Dec 2009, 21:59
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DS Questions- (i got the answer but not sure how that is correct)
1. Are both x and y positive?
i) 2x-2y=1
ii)x/y>1
2. If n and y are both positive integers. 450y=n^3.
Which of the following must be an integer?
i) y/(3x2^2x5) ii) y/(3^2x2x5) iii) y/(3x2x5^2)
a) none
b) I only
c) II only
d) III only
e) I, II & III
3. x, y z are integers greater than 1, what is the value of x+y+z?
i) xyz=70
ii) x/yz=7/10
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Re: GMAT prep quant! i need help [#permalink]
### Show Tags
22 Dec 2009, 23:36
Statement 1 can be verified by picking numbers. From the below table it is clear that Statement 1 is not sufficient.
x Y 2x-2y=1 x>0 & Y>0?
1 ½ 2-1=1 Yes
-1/2 -1 -1+2=1 No
Statement 2 can verified similarly with numbers. Statement 2 is not sufficient.
x Y x/y>1 x>0 & Y>0?
1 ½ 2 > 1 Yes
-1 -1/2 2>1 No
I think taken together also not sufficient.
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Re: GMAT prep quant! i need help [#permalink]
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22 Dec 2009, 23:49
Statement 1 can be verified by picking numbers. From the below table it is clear that Statement 1 is not sufficient.
x Y 2x-2y=1 x>0 & Y>0?
1 ½ 2-1=1 Yes
-1/2 -1 -1+2=1 No
Statement 2 can verified similarly with numbers. Statement 2 is not sufficient.
x Y x/y>1 x>0 & Y>0?
1 ½ 2 > 1 Yes
-1 -1/2 2>1 No
I think taken together also not sufficient.
statement 1 and 2 alone are insuff but together they suffice
from statement1 we have x= (1 + 2y)/2 and from stmnt2 we have x/y >1 . Substituting value of x obtained in stmnt1 in stmnt 2 we have (1 + 2y)/2y > 1 ==> 1/2y + 1> 1 or 1/2y > 0 so y is +ve and for any +ve value of y x will also be +ve.
Hence C
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Re: GMAT prep quant! i need help [#permalink]
### Show Tags
22 Dec 2009, 23:54
bidishabarpujari wrote:
3. x, y z are integers greater than 1, what is the value of x+y+z?
i) xyz=70
ii) x/yz=7/10
3. will go with A
i. xyz= 70 and all are >1 so only values satisfying this is 2*5*7 so we can find the value. hence suff
ii. x/yz = 7/10 ==> 10x = 7 yz => we can have a lot of possibilities here.
e.g.
x = 7 and y and z = 2 and 5
x= 70 and y and z can be (5,20) (10,10)
hence insuff
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Re: GMAT prep quant! i need help [#permalink]
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23 Dec 2009, 00:03
bidishabarpujari wrote:
DS Questions- (i got the answer but not sure how that is correct)
2. If n and y are both positive integers. 450y=n^3.
Which of the following must be an integer?
i) y/(3x2^2x5) ii) y/(3^2x2x5) iii) y/(3x2x5^2)
a) none
b) I only
c) II only
d) III only
e) I, II & III
b) I only
On factorising 450 we get 2* 3^2 * 5^2
so we can write 450y = n^3 as 2* 3^2 * 5^2 y = n^3
Minimum value of y which will satisfy this equation will be 2^2*3*5
only option I y/ 2^2*3*5 will give an integer always
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Re: GMAT prep quant! i need help [#permalink]
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23 Dec 2009, 08:07
The OA of Q1 is c, Q2 is b and Q3 is a.
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Re: GMAT prep quant! i need help [#permalink]
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23 Dec 2009, 09:26
Thanks KP for clarifying.
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Re: GMAT prep quant! i need help [#permalink]
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23 Dec 2009, 09:40
Thanks KP for clarifying.
anytime mate
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Re: GMAT prep quant! i need help [#permalink] 23 Dec 2009, 09:40
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I'm a full time working dad that tries to keep up with technology. I want to haev this blog to share about my life, my journey, places I visit, lifestyle, technology, beauty, business and other topics. I hope you enjoy reading it.
# Royal Pitch
Information From Around The Globe
# How Many Minutes Are In 3 Days
To know how many minutes are in 3 days, you should first know how many hours are in a day. Then, divide that number by 120, which is equal to 3,260 minutes. Similarly, three days are about 1,440 hours. Then, divide it by 360 to find the number of minutes in a day. Lastly, multiply this answer by the number of hours in a day and you will have the answer in seconds.
The answer to the question “How many minutes are in 3 days?” can be found by considering that three days equal 86,400 seconds. This is enough to know how long you’ve been awake and how long you’ve been asleep. For this task, you should divide your total duration into two days, one each hour. Each day contains a certain number of seconds. So, if you’re interested in knowing the duration of a day, you should know the answer to the following question:
The answer to the question “How many minutes are in 3 days?” is a simple calculation. The first step is to look up the unit of time in a calendar. A day consists of 24 hours. By dividing 3 days by 24, you will have 72 hours. The second step is to convert the value to seconds. In other words, a minute is the same as an hour. So, if you need to know how many minutes are in a day, you should divide your entire working day by three.
Likewise, a day is a unit of time, which can be expressed in seconds, minutes, or hours. The average length of a day is 86,400 seconds. Thus, three days are equivalent to around 3,070 minutes. It is not possible to calculate the exact value in nanoseconds, milliseconds, or microseconds, but it’s possible to find the answer using a calculator.
The answer is 3,760 minutes. Then, you can add two more hours and minutes to the previous. This way, you can calculate the time in the future. Once you know the time in the future, you can use the days and hours calculator. The three-day interval is equal to 940 seconds. Therefore, a day is equivalent to three hours and a minute is equal to 480 seconds. So, you can even find the time in the future by using this formula.
How many minutes are in 3 days? There are 72 hours in a day. To calculate the time in minutes, simply multiply 3 days by 24. If you want to measure time in nanoseconds, you should multiply the value by ten. Then, you can estimate the time in microseconds. In a week, you can make the conversion from two hours to a full day. If you need to know the time in milliseconds, you need to use the daily to the minute.
In fact, the answer to how many minutes are in 3 days is quite simple. If you are looking for a more complicated answer, you should use a calculator that is built on the days. This way, you can estimate the time in the minutes and hours. It will also be possible to calculate the time in milliseconds. This is a useful tool for learning about the time in different units. So, what is in a day?
During a week, you can spend approximately 86.400 seconds. During this time, you can count on three hours. If you are in school, the average student will have a different workday. So, a half-hour is the equivalent of a quarter, while a third hour is half an hour. So, you can see, there are more minutes in a day than in a week.
If you want to calculate the time in days, you should first know how much time a day is. A day is a whole 1440 minutes, but it is not always clear how many hours a week is in a week. The answer is: a half-day. And a three-day week, a half-day. So, a third-day is a day. So, a month is a quarter-hour.
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#### What's Included
• Smart Notebooks Presentation
• Interactive Excel File
• Activ Inspire Flipchart
• Lesson Plan
• Microsoft PowerPoint Presentation
• Differentiated Worksheet
To find the nth term of a quadratic sequence students are taught to break down the formula into into quadratic, linear and constant components using a table. An interactive whiteboard resource can be used to generate random questions with their solutions for additional practise and consolidation.
##### Differentiated Learning Objectives
• All students should be able to derive the formula for a quadratic sequence in the form n2+c
• Most students should be able to derive the formula for a quadratic sequence in the form an2+c
• Some students should be able to derive the formula for a quadratic sequence in the form an2+bn+c
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There are three common ways to organise data that fall into multiple sets: two-way tables, frequency diagrams and Venn diagrams. Having blogged about frequency diagrams before I thought I would write about how to draw a Venn Diagram to calculate probabilities. Recapping Two-Way Tables This activity works well to review two-way tables from the previous […]
#### Calculations with Percentages
Students learn how to find a percentage of an amount using calculator and non-calculator methods. As learning progresses they use decimal multipliers to find a percentage change and calculate a simple interest in financial mathematics. This topic follows on from Fractions, Decimals and Percentages and takes place in Year 8 Term 5. Calculations with Percentages […]
#### Proving Geometrical Relationships using Algebra
Back in May 2017 maths teachers around the country eagerly awaited the first exam for the new GCSE Mathematics syllabus. Proving geometrical relationships using algebra featured at grade 9. In Paper 1 of Edexcel’s test paper the last question of the higher tier looked like this. Edexcel wrote about student’s performance on this question in […]
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# College Algebra with Trigonometry (9th Edition) View more editions Solutions for Chapter 7
• 2265 step-by-step solutions
• Solved by professors & experts
• iOS, Android, & web
Chapter: Problem:
In solving certain kinds of more advanced applied mathematical problems—problems dealing with electrical circuits, spring- mass systems, heat flow, and so on—the solution process leads naturally to a function of the form
Functions like this are often easier to work with if transformed into the more familiar form
The process of finding A, B, and C, given M, N, and B, requires a little ingenuity and the use of the sum identity
How do we proceed? We start by trying to get the right side of equation (1) to look like the right side of identity (3). Then we use equation (3), from right to left, to obtain equation (2).
(A) Establishing a Transformation Identity. Show that
where C is any angle (in radians if t is real) having P − (M, N) on its terminal side. [Hint: A first step is the following:
(B) Use of Transformation Identity. Use equation (4) to transform
into the form y2 = A sin (Bt + C), where C is chosen so that |C| is minimum. Compute C to three decimal places. From the new equation, determine the amplitude, period, and phase shift.
(C) Graphing Calculator Visualization and Verification. Graph y1 and y2 from part C in the same viewing window.
(D) Physics Application. A weight suspended from a spring, with spring constant 64, is pulled 4 centimeters below its equilibrium position and is then given a downward thrust to produce an initial downward velocity of 24 centimeters per second. In more advanced mathematics (differential equations) the equation of motion (neglecting air resistance and friction) is found to be given approximately by
where y1 is the coordinate of the bottom of the weight in Figure 1 at time t ( y is in centimeters and t is in seconds). Transform the equation into the form
and indicate the amplitude, period, and phase shift of the motion. Choose the least positive C and keep A positive.
(E) Graphing Calculator Visualization and Verification. Graphy1 and y2 from part E in the same viewing window of a graphing calculator, 0 ≤ t ≤ 6. How many times will the bottom of the weight pass y = 2 in the first 6 seconds?
(F) Solving a Trigonometric Equation. How long, to three decimal places, will it take the bottom of the weight to reach y = 2 for the first time?
Figure 1 Spring-mass system.
Sample Solution
Chapter: Problem:
• Step 1 of 6
(A) y = M sin Bt + N cos Bt
y = (M sin Bt + N cos Bt)
y =
Now, if C has (M, N) on its terminal side,
cos C = sin C =
Thus y = (cos C sin Bt + sin C cos Bt)
y = sin(Bt + C) as required
• Step 2 of 6
(B) A = = = 5
B =
cos C = sin C = C is a 2nd quadrant angle: C = cos–1 = 2.498
y2 = 5 sin
Amplitude = 5, Period = = 4π, Phase shift = – = 4.996
• Step 3 of 6
(C)
• Step 4 of 6
(D) A = = 5
B = 8
cos C = – sin C= – C is a third–quadrant angle: C = π + sin–1 = 4.069
y2 = 5 sin(8t + 4.069)
Amplitude = 5 Period = = Phase shift = – = –0.509
• Step 5 of 6
(E) Graphing y1, y2 and y3 = 2:
The bottom of the weight will pass y = 2 15 times in the first 6 seconds.
• Step 6 of 6
(F) Solve
5 sin(8t + 4.069) = 2
sin(8t + 4.069) = 0.4
8t + 4.069 = sin–1 0.4 or – sin–1 0.4
t = or
t = –0.457 or –0.364
Adding multiples of period : t = 0.328 is the first positive solution, confirmed by the calculator intersection routine.
Corresponding Textbook
College Algebra with Trigonometry | 9th Edition
9780073519500ISBN-13: 0073519502ISBN: Raymond A. BarnettAuthors:
Alternate ISBN: 9780071221757, 9780077297244, 9780077350109, 9780077417864, 9780077941840, 9780077988364
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# Mathematics Paper 1 - 2021 KCSE Prediction Questions and Answers Set 1
### Instructions to Candidates
1. This paper consists of two sections: Section I and Section II.
2. Answer ALL questions from section I and ANY FIVE from section II
4. Non – Programmable silent electronic calculators and KNEC mathematical tables may be used, except where stated otherwise.
SECTION I (50 MARKS)
Answer ALL the questions from this section.
1. Find the reciprocals of 0.216 correct to 3 decimal places hence evaluate; (3 marks)
2. Without using a calculator, evaluate; (4 marks)
3. Three bells ring at intervals of 9 minutes, 15 minutes and 21 minutes. The bells will next ring together at 11.00pm. Find the time the bells had last rang together. (3 marks)
4. Simplify the expression. (3 marks)
5. Without using a calculator or mathematical tables, solve the equation. (4 marks)
2log10x−3log102+log1032=2
6. A line L passes through point (3, 1) and is perpendicular to the line 2y = 4x + 5.
Determine the equation of line L. (3 marks)
1. A forex bureau in Kenya buys and sells foreign currencies as shown below.
Buying Selling Currency Ksh Ksh Chinese Yuan 12.34 12.35 South African rand 11.28 11.37
A businesswoman from China converted 195,250 Chinese Yuan into Kenya shillings.
1. Calculate the amount of money in Kenya shillings that she received. (1 mark)
2. While in Kenya, the businesswoman spent Ksh.1,258,000 and then converted the balance into South African Rand. Calculate the amount of money to the nearest Rand that she received. (3 marks)
2. The seventh term of an A.P is 20 and the sum of the first 20 terms is 610. Find the first term and the common difference. (3 marks)
3. The points P, Q and R lie on a straight line. The position vectors of P and R are 2i + 3j + 13k and 5i – 3j + 4k respectively. Q divides PR internally in the ratio 2 : 1.
Find
1. The position vector of Q. (2 marks)
2. Distance of Q from the origin. (1 mark)
4. In the figure below, <MNO = 54o, <PLM = 50o, PN = NM and PO is parallel to LM.
Find the value of <LPM.
(3 marks)
1. Two matrices A and B are such that
and Given that the determinant of AB = 10, find the value of k. (3 marks)
2. Two pipes A and B can fill an empty tank in 3 hours and 5 hours respectively. Pipe C can empty the full tank in 4 hours. If the three pipes A, B and C are opened at the same time, find how long they will take to fill the tank. (3 marks)
3. Determine the equation of the normal to the curve y = x2 – 3x + 1 at a point (−2, 3) giving your answer in the form ax + by = c. (3 marks)
4. The size of each interior angle of a regular polygon is five times the size of exterior angle. Find the number of sides of the regular polygon. (3 marks)
5. Find the integral values of x which satisfy the inequalities.
x + 8>4x – 6≥3(4 – x) (2 marks)
6. John had two bags A and B containing sugar. If he removed 2kg from bag A and added to bag B, the mass of sugar in bag B would be four times the mass of sugar in bag A. If he added 10kg of sugar to the original amount of sugar in each bag, the mass of sugar in B would be twice the mass of the sugar in bag A.
Calculate the original mass of sugar in each bag. (3 marks)
SECTION II (50 MARKS)
Answer any FIVE questions from this section
1. The table below shows the height measured to the nearest cm of 101 pawpaw trees.
Height in cm 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 Frequency 2 15 18 25 30 6 3 2
1. State the modal class. (1 mark)
2. Calculate to 2 decimal places.
1. The mean height (4 marks)
2. The difference between the mean height and the median height. (5 marks)
1. Three pegs R, S and T are on the vertices of a triangular plan field. R is 300m from S on a bearing of 300o and T is 450m directly South of R.
1. Using a scale of 1cm to represent 60m, draw a diagram to show the position of the pegs. (3 marks)
2. Use the scale drawing to determine;
1. The distance between T and S in metres. (2 marks)
2. The bearing of T from S. (1 mark)
3. Find the area of the field in hectares correct to one decimal place. (4 marks)
2. Mr. Mutuku owns a bicycle which he sometimes rides to go to work. Out of 21 working days in a month he only rides to work for 18 days. If he rides to work the probability that he is bitten by a rabid dog is 4/15 otherwise it is 1/13 only. When he is bitten by the dog, the probability that he will get treatment is 4/5 and if he does not get treatment the probability that he will get rabies is 5/7
1. Draw a tree diagram to show the events. (3 marks)
2. Using the tree diagram above determine the probability that
1. Mutuku will not be bitten by a rabid dog. (2 marks)
2. He will get rabies. (2 marks)
3. He will not get rabies. (3 marks)
1. The volume Vcm3 of a solid depends partly on r2 and partly on r3 where r is one of the dimensions of the solid. When r = 1, the volume is 54.6cm3 and when r = 2, the volume is 226.8cm3.
1. Find the expression for V in terms of r. (5 marks)
2. Calculate the volume of solid when r = 4. (3 marks)
3. Find value of r for which the two parts of the volume are equal. (2 marks)
1. The equation of a curve is given by y = 5x−x2
1. Draw the curve of y = 5x−1/2xfor 0≤x≤6 (3 marks)
2. By integration find the area bounded by the curve, the line x = 6 and the x-axis. (3 marks)
3.
1. On the same graph in (a) draw the line y = 2x (1 mark)
2. Determine the area bounded by the curve and the line y = 2x (3 marks)
1. The diagram below shows a cross section of a bottle. The lower part ABC is a hemisphere of radius 5.2cm and the upper part is a frustum of a cone. The top radius of the frustum is one third of the radius of the hemisphere. The hemisphere part is completely filled with water as shown in the diagram.
When the container is inverted, the water now completely fills on the frustum part.
1. Determine the height of the frustum part. (7 marks)
2. Find the surface area of the frustum part of the bottle. (3 marks)
1. A bus left Nairobi at 7.00a.m and travelled towards Eldoret at average speed of 80km/h. At 7.45am, a car left Eldoret towards Nairobi at an average speed of 120km/hr. The distance between Nairobi and Eldoret is 300km. Calculate
1. The time the bus arrives at Eldoret. (2 marks)
2. The time of the day the two vehicles met. (4 marks)
3. The distance from Nairobi to where the two vehicles met. (2 marks)
4. The distance of the bus from Eldoret when the car arrived at Nairobi. (2 marks)
1.
1. Complete the table below for the function y = 2x3 + 5x2 – x – 6. (3 marks)
x −4 −3 −2 −1 0 1 2 2x3 −128 −54 5x2 −x 4 3 −6 −6 −6 −6 −6 −6 −6 −6 y −50 −6 0
2. Draw the graph of y = 2x3 + 5x2 – x – 6 for (3 marks)
3. By drawing a suitable line, use the graph in (b) to solve the equations
1. 2x3 + 5x2 + x – 4 = 0 (2 marks)
2. 2x3 + 5x2 – x + 2 = 0 (2 marks)
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## how to calculate 1:2:4 ratio concrete cement, sand, metal
DETAILS: cement = 1, sand = 2 metal = 4 so wat is the ratio we want mixing (1:2:4)=1+2+4=7 7 is mixing propotion volume of wet cement conceret is 154 to 157 unit weigth of cement is 1440kg/cum one bag of cement =50/1440=0034722 cum solution : cement = 154/7=022 022/003472=633 bags one baghow many bags of cement are in ratio 1 2 4,now come to the ratio of concrete which is 1:2:4 so sum of the ratio will be 1+2+4=7 calculating separately each material: Cement: 1/7x154 = 22 cubic feet where 1 bag of cement measured 125 cubic feet approx therefore 22/125 = 176 bags of cement 1 bag contains 50kg cement so total cement by weight we need for 100 cubic feet concrete = 17 For 1:2:4 ratio the weight of cement is 50 kg then how ,It is the ratio of cement:fine aggregate (iesand) :coarse aggregate So , for the ratio 1:2:4 If you take 50 kg of cement, Then, 2*50 = 100kg of sand And 4*50 = 200kg of coarse aggregateHow to Calculate Quantities of Cement, Sand and Aggregate ,In the step 3 of “How To Calculate Quantities Of Cement, Sand And Aggregate For Nominal Concrete Mix (1:2:4)” you have calculated that: 01 cum of concrete will require Cement required = 1/0167 = 598 Bags ~ 6 Bags Sand required = 115/0167 = 688 Kgs or 1498 cfthow many bags of cement are in ratio 1:2:4,How many 25Kg bags of cement do I need for 1 cubic metre of all-in-ballast when mixing concrete at a ratio of 1:6 Asked by: anon-180810193004282405 14th Aug 2011 4 Answers Best AnswerHow to calculate the cement quantity with a ratio of 1:2:4 ,I furnish below quantities of materials required 100cubicfeet of ratio 1:2:4:-3/4” size hard broken stone 90 cubicfeet Dry river sand 45 cubicfeet Cement 18 bags of 50 kg each For your calculated volume of concrete, you may work out the cement on pro rats basis For example your volume ofcalculation of cement bags in 1:2:4 concrete mix,calculation of cement bags in 1:2:4 concrete mix Answer / smohamed imthiyaz 1+2+4=7 then total ingredient=157 1part of cement So 1 / 7 =0142 Then 0142*157=02229 Unit weight of concrete is 1440kg/m3 02229*1440=32097kg 1cement bag=50kg 32097/50= 641 bags Is This Answer Correct ? How many bags of cement will be used for one cubic meters ,How many bags of cement will be used for one cubic meters using ratio 1:2:4 mix Products As a leading global manufacturer of crushing, grinding and mining equipments, we offer advanced, reasonable solutions for any size-reduction requirements including, How many bags of cement will be used for one cubic meters using ratio 1:2:4 mix, quarry, aggregate, and different kinds of mineralsConcrete Calculator - Estimate Cement, Sand, Gravel ,Example calculation Estimate the quantity of cement, sand and stone aggregate required for 1 cubic meter of 1:2:4 concrete mix Ans Materials required are 7 nos of 50 kg bag of cement, 042 m 3 of sand and 083 m 3 of stone aggregateCalculate cement sand and aggregate - Civil RnD,Concrete mix ratios are usually in the form of cement : Sand: Aggregate For example, if the concrete mix ratio of M20 concrete is 1:15:3 then 1 part of cement, 15 part of sand and 3 part of aggregate in volume should be taken to produce concrete Prescribed concrete Mix ratios for all grades of concrete
## How many bags of cement will be used for one cubic meters
How many bags of cement will be used for one cubic meters using ratio 1:2:4 mix Products As a leading global manufacturer of crushing, grinding and mining equipments, we offer advanced, reasonable solutions for any size-reduction requirements including, How many bags of cement will be used for one cubic meters using ratio 1:2:4 mix, quarry, aggregate, and different kinds of mineralsHow To Calculate Quantities Of Cement, Sand And Aggregate ,As the latest generation of 53 grade OPC cement is ultimately giving a strength of 65 to 70 MPa at 28days, 1:2:4 will give a strength of M20 A detail procedure to calculate the cement bags required for 1: 2 :4 mix (~6 bags of cement per cum) is shown belowHow Many Bags of Cement Are in a Cubic Yard? | Hunker,Mixing Concrete by the Pound You'll need to use the weight of one bag of cement to calibrate your "part" measurements For instance, if you're using 80-pound bags of cement, one "part" equals 80 pounds Thus, for a standard mix using 80-pound bags, you'll need 40 pounds of water (80 x 1/2 = 40), 160 pounds of dry sand (80 x 2 = 160) and 80 pounds of crushed stone aggregateHow To Calculate Cement Bags In 1 Cubic Meter - Daily Civil,How To Calculate Cement Bags In 1 Cubic Meter? Let us consider the nominal mix is 1:2:4 Wastage of cement during handling is considered as 2% Sum of the ratio = 1+2+4=7 Total material =152 Now 152*(1/7)=02171 Here 1 is the ratio of cement FAHAD khan January 22, 2017 ReplyMaking Concrete - A Layman's Guide to Clean Water,Sometimes, a different cement:sand:gravel ratio such as 1:2:4 or 1:2:2 is printed on a sack of cement as a guideline for making concrete slabs It is best to use the ratio printed on the sack, but ifHow to calculate the numbers of bags of cement in the ,Now, for the plan above, the numbers of bags of cement needed in the concrete is simply calculated by dividing the total volume of cement in concrete by a volume of one bag which is taken as 003 For total volume of cement in ratio 1:2:4, add the numbers together,1+2+4=7 Divide the volume of concrete by 7 to get the volume of cement in concreteQuantity of Cement and Sand Calculation in Mortar,Quantity of Cement and Sand Calculation in Mortar Quantity of cement mortar is required for rate analysis of brickwork and plaster or estimation of Since the volume of 1 bag of cement is 00347 m 3, so the number of bag of cement will be calculated as: Example: For cement mortar of 1:6, the quantity calculated will be as below: how many bags of 40kg cement do you need in a 1 cubic ,Jul 21, 2008 · Best Answer: First find the yield from one bag say water cement ratio is 04 40 kg of cement 80 kg fine aggregate (2 x 40) 240kg of coarse aggregate (3 x 40) 16kg water Total = 376kg Say 1m3 = 2400kg so 2400/ 376 = 64 bags This is a simple method and you would need to know actual densities etc of the CONCRETE MIXING RATIOS - Everything About Concrete,,A concrete mixture ratio of 1 part cement, 3 parts sand, and 3 parts aggregate will produce a concrete mix of approximately 3000 psi Mixing water with the cement, sand, and stone will form a paste that will bind the materials together until the mix hardensConcrete Calculator - How Much Do I Need? | QUIKRETE ,CONCRETE CALCULATOR - How Much Do I Need? You can use this concrete calculator to help you determine the number of bags of QUIKRETE® Concrete Mix, Mortar Mix, or Fast-Setting Concrete you will need for the following projects
## How Many Cubic Feet Do You Get Per Bag Of Portland Cement
Aug 15, 2017 · "How Many Cubic Feet Do You Get Per Bag Of Portland Cement? Watch more videos for more knowledge How Many Cubic Feet Do You Get Per Bag Of https://youtube What's a 1:2:4 Mix?| Concrete Construction Magazine | Mix ,If the proportions are based on weight, use 4 pounds of coarse aggregate and 2 pounds of sand for every pound of cement To make the concrete, though, you still need to know the required water-cement ratio If the proportions are based on volume, use four shovelfuls of coarse aggregate and two shovelfuls of sand for every shovelful of cementHow Many Bags Of Concrete Do I Need: How - Housesumo,How Many Bags Of Concrete Do I Need: How Much Concrete Do I Need Concrete Bags Regular concrete mix sold at lumber yards and home centers comes in different sized bagsWhat is 5 bag mix, 6 bag mix, 3000 PSI, 3500 PSI, 4000 PSI ,What is 5 bag mix, 6 bag mix, 3000 PSI, 3500 PSI, 4000 PSI, etc… As a Chicago Concrete Contractor serving Chicago, IL and all surrounding suburbs, I always find myself helping our clients become more educated on the concrete they will be receivingcement concrete ratio 1 2 4 calculation in 100 cft,how to calculate 1:2:4 ratio concrete cement, sand, metal ratio is 1:2:4 sum is equle 7 1 bag cement is equle to 125 total quantity resum is 200 cft 1m3 of 1:2:4 nominal concrete mix contains cement634 bags sand 055 m3 with 25% bulking in normal practice metal 088m3How to Calculate Cement, Sand Quantity for Plastering ,Definitive Guide of Plastering work ratio (cement and Sand) calculation with crystal clear explanation and also with easy go to plastering cement calculator How to calculate cement, sand quantity for Plastering? Before beginning to work on the plastering calculation, note down these general things = 766 bags (Approx – 8 Bags) Sand how many bags of 40kg cement do you need in a 1 cubic ,Jul 21, 2008 · Best Answer: First find the yield from one bag say water cement ratio is 04 40 kg of cement 80 kg fine aggregate (2 x 40) 240kg of coarse aggregate (3 x 40) 16kg water Total = 376kg Say 1m3 = 2400kg so 2400/ 376 = 64 bags This is a simple method and you would need to know actual densities etc of the how we calculate of Sand, cement and aggregate of M20 ,how we calculate of Sand, cement and aggregate of M20 ratio or other ratio? Ex 1:15:3 is the ration of M20 and now how much sand or cement or aggregate we use in this? Civil EngineeringHow many Bags of 50Kg cement would be used for 1Cum of ,How many Bags of 50Kg cement would be used for 1Cum of concrete ratio 1:2:4? Civil Engineering Question added by Oluwasogo Faleye , Quantity Surveyor , Structuracasa Nig LtdWhat is the quantity of cement required for 100 sqm ,thickness 4cm 0 04m volume of wet concrete is1 54 to1 57 ratio of mix1 2 4 volume 1 2 4 7 unit weight of cement bag is1440 kg cum one bag of cement 50
## How Many Bags Of Concrete Do I Need: How - Housesumo
60-pound bags yield 045 cubic feet and 40-pound bags just 030 cubic feet The actual yield is approximate because the amount of water added to the mix may vary 60 to 80 pounds is a lot of weight to lift, carry and handle If you don’t feel up to that, you can buy the smaller bags insteadHOW MANY BAG OF CEMENT IN 1 CUBIC METRE - Facebook,HOW MANY BAG OF CEMENT IN 1 CUBIC METRE OF CONCRETE How many bag o f cement are in 1 cum concrete ratio 1:2:4 with 20mm aggregate, water-cement ratio 05? SOLUTION Total ratio =1+2+4=7 CEMENT % of cement in 1m2 of concrete = 1/7 = 0143 Add 50% waste =0072 TOTAL =0215 In 1 m2 of cement we have 288bags of cement Therefore % of cement in 1 m2 of concrete = 0215 x 288 =How many bags of cement do you have in 1 cubic meter of ,Sep 13, 2006 · Best Answer: 1 bag of cement contains 1ft^3 divide 1 cubic meter into 13 parts 1 part cement, 4 parts fine aggregate and 8 parts course aggregate 1cubic meters of concrete/13= 00769 cubic meters Since cement is only 1 part then you will need 00769 cubic meters of cement 00769m^3 ( 339ft/1m)^3= 2996 ft^3 of cementHow to Calculate the Water to Cement Ratio - The Concrete ,The water to cement ratio is calculated by dividing the water in one cubic yard of the mix ( in pounds) by the cement in the mix (in pounds) So if one cubic yard of the mix has 235 pounds of water and 470 pounds of cement- the mix is a 50 water to cement ratioCONCRETE RULES OF THUMBS,CONCRETE RULES OF THUMBS BLOCK 3 Bags Mortar per 100 Block (35 Block per Bag) 1000 lb Sand per 100 Block 1125 Block per sq ft of Wall Area 75% of Wall Length give Block per Course Number of Courses = 15 X Height of wall Glass Block 25-8x8’s per Bag Mortar Brick weight (average) 356lbs std size 435lbs oversized BRICKHow Many Bags of cement ,sand,Crush in ratio of 1;2:4 ,Sep 08, 2008 · Want to erect a post in a pit size 150 M wide x 1050m long and 0975M deep , want to use concretect in the ration of 1:2:4, can anyone suggest, how many bags of Cement, how much sand and concrete in Cum will be requierdhow many bags of 50 kg of cement required for 100 cubic ,May 18, 2013 · Best Answer: I guess you mean 100 cubic feet of concrete, in which the ratio of cement:sand:aggregate is 1:2:4 According to the Portland Cement Association, the volume of a 94-lb bag of cement may be taken to be 1 cubic foot, BUT the final volume of concrete will be only 2/3 of the sum of the volumes of the componentsHow To Mix Concrete - Easy Step By Step Instructions For ,A bag of 80 lb quikcrete sells for about \$335 US dollars in Maine A bag of 94 lb portland cement sells for \$863 US dollars in Maine A bag of 50 lb sand sells for \$234 US dollars in Maine Learning how to mix concrete by hand is simple but it is hard work,,
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Quadratic Formula Calculator; GCF Calculator; LCM Calculator; cosine calculator; sine calculator; inverse cosine calculator; Area of Triangle calculator; Related Articles. Related Calculators. Get homework help now! is defined as the ratio of the adjacent leg to the hypotenuse, or . Enter the cosine value, select degrees (°) or radians (rad) and press the = button: cos-1 Calculate: Result: Math Calculators General Math. The inverse sine y=sin^(-1)(x) or y=asin(x) or y=arcsin(x) is such a function that sin(y)=x. This thread is locked. There are 2 different ways that you can enter input into our arc tan calculator. Scientific Calculator. You can enter input as either a decimal or as the opposite over the adjacent. The cosine calculator allows through the cos function to calculate online the cosine of an angle in radians, you must first select the desired unit by clicking on the options button calculation module. The domain of the inverse sine is [-1,1], the range is [-pi/2,pi/2]. to find missing angles and sides if you know any 3 of the sides or angles. Am I able to use a function on Windows 10 Calculator to find the inverse function of sine or cos or tan? The following formula can be used to calculate an angle from any cosine value. In this article. The above output shows that the degree is 180 after conversion. This calculator uses the Law of Sines: $~~ \frac{\sin\alpha}{a} = \frac{\cos\beta}{b} = \frac{cos\gamma}{c}~~$ and the Law of Cosines: $~~ c^2 = a^2 + b^2 - 2ab \cos\gamma ~~$ to solve oblique triangle i.e. Fourier transform calculator. By Yang Kuang, Elleyne Kase . arccos(x) = cos-1(x) Code to add this calci to your website Formula: ACos = Cos-1 ( X ) Example. If you want to convert the result to a different angel use our online angle converter. Inverse cosine calculator. After that, you can start your calculations. Enter the cosine value, select degrees (°) or radians (rad) and press the = button. Calculator Use. Inverse cosine calculator. 1 = ACOS (-1) * 180 / PI Output. The term Arccos is shortform of the term 'Arc Cosine'. Also, the calculator will show you a step by step explanation. The arccos (0.14) is 1.4303 radians and 81.95 degrees. You can access the intrinsic math functions by adding Imports System.Math to your file or project. The following table shows non-intrinsic math functions that can be derived from the intrinsic math functions of the System.Math object. On the calculator you press one of the following (depending on your brand of calculator): either '2ndF sin' or 'shift sin'. It is normally represented by arccos(θ) or cos-1 (θ). Inverse tangent calculator.Enter the tangent value, select degrees (°) or radians (rad) and press the = button. Formula: arccos(y) = cos-1 (y) Firstly, let’s take some examples to understand the process/calculation and formula. They are mirror images (about the diagonal) But why does Inverse Cosine get chopped off at top and bottom (the dots are not really part of the function) ... ? An Easy to use online calculator to calculate Online arccos(x) calculator. Arccos Calculator – Online Inverse Cosine Calculator. In order to calculate the unknown values you must enter 3 known values. As functions of a complex variable, inverse hyperbolic functions are multivalued functions that are analytic, except at a finite number of points. Type in 12+2*3 (=18) Select "deg", type in cos(45) (=0.7071067811865476) Type in 2/sqrt(2) (=1.414213562373095) Function Reference arccos(x) = cos-1 (x) For example, If the cosine of 60° is 0.5: cos(60°) = 0.5. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. The inverse of the cosine is the arccosine function: acos(x) or arccos(x), which takes values between 0 and 180 degrees. … Defination: Arccosine defines as the inverse function of cos(y). Cosine calculator is a Triangle Calculator. To calculate any angle, A, B or C, enter 3 side lengths a, b and c. This is the same calculation as Side-Side-Side (SSS) Theorem. This MATLAB function returns the inverse cosine (cos-1) of the elements of X in degrees. See the below example gives the output in degree. Arccosine(x) = C. Cosine C = x. An online calculator to calculate the inverse cosine function arccos(x) in radians and degrees.. How to use the arccos (x) calculator 1 - Enter x as a real number, within the domain of arccos function such that -1 ≤ x ≤ 1 and the number of decimal places desired then press "enter". Inverse Hyperbolic Cosine (arcosh) calculator online. Uses the law of cosines to calculate unknown angles or sides of a triangle. This is a very powerful Scientific Calculator You can use it like a normal calculator, or you can type formulas like (3+7^2)*2 It has many functions you can type in . Inverse cosine calculator. Arccos Calculator. FREE online Tutoring on Thursday nights! To evaluate inverse trigonometric functions that do not involve the special angles discussed previously, we will need to use a calculator or other type of technology. Calculate Inverse Cosine in Degree with MS Excel. Try an Arccos Calculator to find the inverse cosine of the Radians and Degrees. The inverse hyperbolic functions are the inverse functions of the hyperbolic functions. One important ratio in right triangles is the cosine. It calculates cos(x) value by using degree or radian value of triangle also you can find inverse cosine value. Arccos / Inverse Cosine. Find a local tutor in you area now! Here is Cosine and Inverse Cosine plotted on the same graph: Cosine and Inverse Cosine . Angle in radians: rad: Calculation: Cosine calculator Arccos definition. Enter the cosine value, select degrees (°) or radians (rad) and press the = button. Calculate arccos(x) in degrees and radians with this trigonometric calculator. Arcsin Calculator. Calculate Arcsine, Arccosine, Arctangent, Arccotangent, Arcsecant and Arccosecant for values of x and get answers in degrees, ratians and pi. Inverse Cosine Formula. The Arccos function allows the calculation of the Arc cosine of a number. It is useful for finding an angle x when cos(x) is known. Graphs for inverse trigonometric functions. 180. Inverse Cosine Calculator. Specifically, the arccos is the inverse of the cosine. sin cos table; Trigonometry Formulas for class 11 (PDF download) Trigonometry formulas for class 10; link to this page by copying the following text Specifically, they are the inverses of the hyperbolic sine, cosine, tangent, cotangent, cosecant and secant functions. NumPy: Calculate inverse sine, cosine, and tangent for all elements in a given array Last update on February 26 2020 08:09:27 (UTC/GMT +8 hours) NumPy Mathematics: Exercise-22 with Solution. Write a NumPy program to calculate inverse sine, inverse cosine, and inverse tangent for all elements in a given array. How to calculate the cosine of an angle? How to Calculate the Cosine of an Angle. Arccos calculator to easily calculate the arc cosine (inverse cosine) function of any number. Cos −1 (x), Arccos Example. The Arccos Calculator is also called the Inverse Cosine Calculator, as it is the inverse of the cos value. Using a Calculator to Evaluate Inverse Trigonometric Functions. In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Trigonometric inverse cosine calculator Value / for example: 0.25, 0.5, -0.1 Between -1 and 1 Calculate What is inverse cosine. Calculate the value of Inverse Hyperbolic Cosine (arcosh) trigonometric function instantly using this tool. It is just the inverse function of cos(x). Use our online Arccos calculator and Inverse cosine calculator (cos-1 calculator). Online arctan(x) calculator. Most scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse sine, cosine, and tangent functions. Examples. Example 1: find the exact value for cos[arccos (-0.5)] If the cosine of 120° is -0.5 which give us cos (120°)=-0.5 then cos[arccos (-0.5)]=-0.5. You can follow the question or vote as helpful, but you cannot reply to this thread. Extended Keyboard; Upload; Examples; Random; Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Then the arccos of 0.5 is 60°: arccos(0.5) = cos-1 (0.5) = 60° Arccos table. To evaluate inverse trigonometric functions that do not involve the special angles discussed previously, we will need to use a calculator or other type of technology. For graph, see graphing calculator. Show Instructions. Sample Solution:- Python Code: import numpy as np x = np.array([-1., 0, 1.]) Free Online Inverse Cosine Calculator works in degrees or radians, plus draws triangle. The calculator will find the inverse sine of the given value in radians and degrees. Because you spend a ton of time in pre-calculus working with trigonometric functions, you need to understand ratios. The cosine of an angle, or . To calculate cosine online of pi/6, enter cos(pi/6), after calculation, the result sqrt(3)/2 is returned. To calculate the output in degree, you have to multiply the default output of the function with 180/PI(). Most scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse sine, cosine, and tangent functions. Calculate the number of years (on board time) that a spacecraft accelerating and decelerating at 1g would take for a round trip to the Andromeda galaxy. You have to use the below-given method to perform the degree conversion. Where AC is arc cosine, which is the same as saving inverse cosine Math Gifs; Algebra; Geometry; Trigonometry; Calculus; Teacher Tools; Learn to Code; Home; Trig Calcs; Inverse Tan; Inverse Tangent Calculator. It is an odd function. The result to a different angel use our online angle converter ( x ) in degrees and radians this! Wolfram 's breakthrough technology & knowledgebase, relied on by millions of students professionals. Or vote as helpful, but you can enter input as either a or... Functions of a triangle hyperbolic sine, cosine, tangent, cotangent cosecant. Cos value arccos of 0.5 is 60°: arccos ( 0.5 ) = C. C... Term 'Arc cosine ' a complex variable, inverse hyperbolic functions specific keys or buttons the! Or sides of a number working with trigonometric functions, you have to use online calculator find! Degrees and radians with this trigonometric calculator function with 180/PI ( ) easily the! Solution: - Python Code: import NumPy as np x = np.array ( [,... Also you can find inverse cosine ) function cosine inverse calculator any number defines as the ratio of the with! 'Arc cosine ' perform the degree is 180 after conversion the result to different. Scientific calculators and calculator-emulating applications have specific keys or buttons for the inverse of the hyperbolic sine, cosine and... Or project the below example gives the output in degree output shows the... Represented by arccos ( x ) want to convert the result to a different angel use online! Function allows the Calculation of the inverse sine, cosine, tangent, cotangent, cosecant and secant.. The System.Math object or vote as helpful, but you can enter input either! & knowledgebase, relied on by millions of students & professionals What is cosine!: arccos ( x ) calculator breakthrough technology & knowledgebase, relied on by millions of &..., except at a finite number of points, the range is [ -1,1 ].! Angle from any cosine value, select degrees ( ° ) or radians rad... Also, the calculator will find the inverse function of cos ( )! Unknown values you must enter 3 known values for finding an angle x cos!: 0.25, 0.5, -0.1 Between -1 and 1 calculate What is inverse cosine calculator arccos definition,. ( x ) calculator, the calculator will find the inverse function cos! 'Arc cosine ' = cos-1 ( θ ) or radians ( rad ) and the... Calculate inverse sine, cosine, and inverse cosine - Python Code import... 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1. ## Three tough problems.
Sorry for such vague title.
Anyway, I am doing some math evaluation test when I ran across these bad boys.
For question #1, I have no idea where to begin.
For question #2, I forgot how to deal with absolute values.
For question #3:
$\displaystyle y= x+1$ So, $\displaystyle y^2 = x^2 + 2x +1$ Sub that into the other equation and we get $\displaystyle \frac{x^2}{4} + \frac{x^2 +2x+1}{9} = 1$
$\displaystyle 9x^2 + 4x^2 + 8x + 4 = 36$
$\displaystyle 13x^2 + 8x - 32 = 0$ but we do not get any nice roots, and all the options for the questions are rational.
If some one can show me how to solve these, I would appreciate it
2. ## Re: Three tough problems.
Hello, sakonpure61
$\displaystyle \text{The two curves }y \:=\:x+1\text{ and }\frac{x^2}{4} + \frac{y^2}{9} \:=\:1$
$\displaystyle \text{ intersect in exactly two points }(a,b)\text{ and }(c,d).$
$\displaystyle \text{Find the value of }a+c.$
$\displaystyle y\:=\: x+1 \quad\Rightarrow\quad y^2 \:=\: x^2 + 2x +1$
Sub that into the other equation and we get .$\displaystyle \frac{x^2}{4} + \frac{x^2 +2x+1}{9} \:=\: 1$
$\displaystyle 9x^2 + 4x^2 + 8x + 4 \:=\: 36 \quad\Rightarrow\quad 13x^2 + 8x - 32 \:=\: 0$
But we do not get any nice roots, and all the options for the question are rational.
I bet your work is correct!
Quadratic Formula: .$\displaystyle x \:=\:\frac{\text{-}8 \pm\sqrt{1728}}{26} \:=\:\frac{\text{-}4\pm12\sqrt{3}}{13}$
The points are:
. . $\displaystyle (a,b) \:=\:\left(\frac{\text{-}4+12\sqrt{3}}{13},\:\frac{9+12\sqrt{3}}{13} \right)\;\text{ and }\;(c,d) \:=\:\left(\frac{\text{-}4-12\sqrt{3}}{13},\;\frac{9-12\sqrt{3}}{13}\right)$
$\displaystyle \text{Now add }a\text{ and }c\!: \;\frac{\text{-}4+12\sqrt{3}}{13} + \frac{\text{-}4 - 12\sqrt{3}}{13} \;=\;\cdots$
3. ## Re: Three tough problems.
Oh! I gave up too soon, thank you for the reply.
4. ## Re: Three tough problems.
Hey Sakon, hope that's a HOCKEY jersey in your avatar!!
5. ## Re: Three tough problems.
hehe, kind of an embarrassing situation, just look the other way ;D .
6. ## Re: Three tough problems.
The answer to 10 is B, and here I will teach you how to cheat lol.
If |b| > a > 0, that requires b>a or b<-a. Hence, it is impossible for b to stand between -a and a.
That's why B is incorrect. 6-x^2 cannot stay between -3 and 3 if |6-x^2| >=3.
7. ## Re: Three tough problems.
For the first question, I am gonna find out the equation for line BC first, and you shall see why.
slope: (0-1)/(1- -3) = -1/4, the line should be y=(-1/4)x+b. Plugging C(1, 0) in, you should get b= 1/4 so the line BC is y=(-1/4)x+(1/4)
Here's the thing: since HA and BC are perpendicular, THE PRODUCT OF THEIR SLOPES will be -1.
Hence, you know that the slope for HA is 4.
Then, y=4x+b for HA. Plugging A(-1, 2) in should get b= 6. Then line HA is y=4x+6.
Since H is the intersection point of BC and HA, that means y=4x+6=(-1/4)x+(1/4).
To solve that, I am gonna multiply both sides by 4: 16x+24 = -x+1 => 17x = -23, x=-23/17, y=4(-23/17) + 6 = 10/17.
A?
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# Solved: Use the Taylor method of order two with h = 0.1 to approximate the solution to y
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## Solution for problem 12 Chapter 5.3
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Problem 12
Use the Taylor method of order two with h = 0.1 to approximate the solution to y = 1 + t sin(ty), 0 t 2, y(0) = 0.
Step-by-Step Solution:
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Integrals with partial fractions In this section we are going to take a look at integrals of rational expressions of polynomials and once again let’s start this section out with an integral that we can already do so we can contrast it with the integrals that we’ll be doing in this section. So, if the numerator is the derivative of the denominator (or a constant multiple of the derivative of the denominator) doing this kind of integral is fairly simple. However, often the numerator isn’t the derivative of the denominator (or a constant multiple). For example, consider the following integral.
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This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. The full step-by-step solution to problem: 12 from chapter: 5.3 was answered by , our top Math solution expert on 03/16/18, 03:30PM. Since the solution to 12 from 5.3 chapter was answered, more than 217 students have viewed the full step-by-step answer. The answer to “Use the Taylor method of order two with h = 0.1 to approximate the solution to y = 1 + t sin(ty), 0 t 2, y(0) = 0.” is broken down into a number of easy to follow steps, and 28 words. Numerical Analysis was written by and is associated to the ISBN: 9780538733519. This full solution covers the following key subjects: . This expansive textbook survival guide covers 73 chapters, and 1135 solutions.
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# Ratio analysis theory-prime
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### Ratio analysis theory-prime
1. 1. CHAPTER Ratio 4Analysis L E A R N I N G OVERVIEW 1. Ratio basicsRatio basics 2. Computing ratiosRatio Analysis compares one figure in one financial a. Short term solvencystatement (say P&L account or Balance Sheet) with b. Long term solvencyanother figure in the same financial statement or in c. Asset managementanother financial statement of the company. d. ProfitabilityA ratio is expressed in the numerator denominator e. Marketformat. Thus the numerator and denominator can beeither from the P&L account or the Balance sheet of 3. Interpreting ratiosthe same company. a. Common size analysisRatios give colour to absolute figures. For example a b. Trend analysisprofit of Rs.100 lakhs means very little to an analyst c. DuPont chartbecause he needs to know what the sales was or what d. Limitationsthe networth was against which the Rs.100 lakhs wasearned. More than the profit, the ratio of profit to salesand the ratio of profit to networth is useful tounderstand the performance of a company. Thus ifprofit grew from Rs 100 lakhs to Rs 125 lakhs, whileit is good, what is more important is how it stacked upagainst the sales achieved or the networth deployed.
3. 3. Ratio Analysis 3A major advantage of looking at current assets and current liabilities is that their bookvalues approximate towards their market values. Often these assets and liabilities do notlive long enough for the two to step out of line.1. Current Ratio: This is the ratio of current assets to current liabilities. Current Assets / Current LiabilitiesBecause current assets are convertible to cash in one year and current liabilities arepayable within one year, the current ratio is an indicator of short term solvency. The unitof measure is “times”. For instance if the current ratio is 1.4 we say that the ratio is 1.4times. It means that current assets are 1.4 times the current liabilities.To a short term lender, including a creditor, a high current ratio is a source of comfort.To the firm, a high current ratio indicates liquidity, but it also may mean inefficient use ofcash and other current assets. A ratio of 1.33 is considered welcome.The current radio is affected by various types of transactions. For example suppose thefirm borrows over the long term to raise money. The short term effect would be anincrease in cash and an increase in long term debt. So the current ratio would rise.Finally, a low current ratio is not necessarily bad for a company which has a largereservoir of untapped borrowing.2. Quick or Acid test Ratio: This is the ratio of quick assets to current liabilities or toquick liabilities. Quick Assets / Current Liabilities Quick Assets / Quick LiabilitiesThree points merit attention.a. Inventory: The book values of inventory are least reliable as measures of realisable value because over time they may become lost, damaged or obsolete. Further, to an external analyst the market value of inventory may not be available since they are carried in the books at cost. Large inventories are often a sign of short-term trouble. The firm may have overestimated sales and consequently may have overbought or overproduced leading to a substantial part of the liquidity locked in low moving inventory. Hence inventory is eliminated from current assets to arrive at quick assets.b. Prepaid expenses. Prepaid expenses too are deducted from current assets since they are not really convertible into cash. They are only adjustments against future payments.c. Overdraft: In practice, overdraft is not exactly repayable within 12 months because it is almost always renewed. Therefore there is a view that in computing quick liabilities we must deduct overdraft from current liabilities.Prime Academy FL in CAFM
4. 4. 4 Ratio Analysis3. Cash Reservoir Ratio: Does a company have enough cash or cash equivalents to meet its current liabilities? The Cash reservoir ratio measures this. Cash Reservoir / Current LiabilitiesCash Reservoir = Cash + Bank + Marketable securities.Alternatively, Cash Reservoir = Current Assets – Inventory.But the former one is more appropriate. A very short term creditor (one who gives money for say a week or 15 days) should be interested in this ratio.B: Capital Structure or Long Term Solvency RatiosLong term solvency ratios measure the firm’s long term ability to meet its paymentobligations. They are also referred to as leverage ratios. Back in the chapter CapitalStructure Planning you learnt about financial leverage as arising out of the existence ofdebt in the capital structure. In Introduction to Financial Management we understood thisas being the first quadrant of the balance sheet.4. Total debt ratio: This is the ratio of total debt to total assets. Total Debt / Total assetsThe term “total debt” means all debt; both long term and short term i.e. it includes currentliabilities. The term “total assets” means all assets; both fixed assets and current assets.There are two variants to this ratio namely debt-equity ratio and equity multiplier. a. The debt equity ratio is measured as total debt to total equity. b. The equity multiplier is the ratio of total assets to total equityThe equity multiplier is 1 plus debt equity ratio. Given any one of these three ratios, youcan immediately compute the other two so they all say the same thing.5. Times interest earned (Interest coverage ratio): This is the ratio of EBIT toInterest. EBIT / InterestThe interest referred to here is the interest on both long term and short term loan. Theratio measures how much earnings are available to cover interest obligations. If coverageis computed only for long term interest then only long term interest should be consideredin the denominator and the EBIT will mean earnings before long term interest and taxes.There are various variants to the above ratio. For instance, there is a view that the earningshould be recorded after tax i.e. earnings before interest but after tax. And that thedenominator will be unchanged at Interest. However we have stuck to the moretraditional and more popular view.Prime Academy FL in CAFM
5. 5. Ratio Analysis 56. Cash coverage: This is the ratio of ‘EBIT plus depreciation’ to Interest. (EBIT + Depreciation ) / Interest Need to compute cash cover While interest is a cash measure, EBIT is not. That’s because it has taken into account depreciation which is a non-cash charge.This ratio is considered as a measure of the firm’s ability to generate cash from operationsand is used as a measure of cash flow available to meet financial obligations.C: Asset Management or Turnover RatiosThe Asset management ratios (a k a Asset turnover ratios) measure the efficiency withwhich a company deploys its assets to generate sales.7. Total Assets turnover ratio: This is the ratio of sales to total assets. Sales / Total AssetsWhile “total assets” is technically more correct, average assets could also be used.Average asset is the simple average of opening and closing assets.If the total assets turnover ratio is 4, it means that for every rupee invested we havegenerated Rs.4 of sales. The term total assets would be the sum of fixed assets andcurrent assets.The higher the ratio the better it is for the company.The reciprocal of the total assets turnover ratio is the “Capital Intensity ratio”. It can beinterpreted as the rupee invested in assets needed to generate Re.1 of sales. High valuescorrespond to capital intensive industries. 1 / Total assets turnover ratioThe total assets turnover ratio can be split into FATO and WCTO ratio.8. Fixed Assets turnover ratio (FATO): This is the ratio of sales to fixed assets.The fixed assets should typically be on net basis i.e. net of accumulated depreciation. Sales / Net fixed assetsAverage fixed assets i.e. the simple average of opening and closing fixed assets can alsobe used.If the fixed assets turnover ratio is 3, it means that for every rupee invested in fixed assetswe have generated Rs.3 of sales.The higher the ratio the better it is for the company.Prime Academy FL in CAFM
6. 6. 6 Ratio Analysis9. Working capital turnover ratio (WCTO): This is the ratio of sales to networking capital. Net working capital would mean current assets less current liabilities. Sales / Net Working CapitalAverage working capital i.e. the simple average of opening and closing working capitalcan also be used.If the working capital turnover ratio is 6, it means that for every rupee invested inworking capital we have generated Rs.6 of sales.The higher the ratio the better it is for the company.This ratio becomes more understandable if we convert it into number of days. If weturned over our working capital 6 times a year, it means that the working capital wasunlocked every 60 days. This is called the working capital days’ ratio and is given bythe following formula: 365 / Working capital turnover ratioThe lower this ratio, the better it is for the company.The working capital turnover ratio can now be broken into its component parts.10. Inventory turnover ratio: This is the ratio of cost of goods sold to closinginventory. Cost of goods sold / InventoryIt can also be expressed as the ratio of cost of goods sold to average inventory. Whileclosing inventory is technically more correct, average inventory could be used since anexternal analyst is unsure whether the year end numbers are dressed up.The numerator is “Cost of goods sold” and not sales because inventory is valued at cost.However to use “Sales” in the numerator is also a practice that many adopt.If the inventory turnover ratio is 3, it means that we sold off the entire inventory thrice.As long as we are not running out of stock and hence losing sales, the higher this ratio is,the more efficient is the management of inventory.If we turned over inventory over 3 times during the year, then we can say that we heldinventory for approximately 121 days before selling it. This is called the average days’sales in Inventory and is given by the following formula: 365 / Inventory turnover ratioThe ratio measures how fast we sold our products. Note that inventory turnover ratio andaverage days’ sales in inventory measure the same thing.11. Receivable / Debtors turnover ratio: This is the ratio of sales to closing debtors. Sales / DebtorsPrime Academy FL in CAFM
7. 7. Ratio Analysis 7While closing debtors is technically more correct, average debtors could be used since anexternal analyst is unsure whether the year end numbers are dressed up.If the debtors’ turnover ratio is 8, it means that we collected our outstanding 8 times ayear. As long as we do not miss out sales, the higher this ratio is, the more efficient is themanagement of debtors.This ratio is far easier to grasp if we converted it into number of days. If we turned overdebtors 8 times a year, we can say that debtors on an average were 45 days. This is calledthe average days’ sales in receivable and is given by the following formula: 365 / Receivable turnover ratioThe ratio is often called the Average Collection period.12. Payables / Creditors turnover ratio: In so far as we wanted to know how wellwe used our debtors we must also know how well we utilise the creditors. Towards thiswe compute the Creditors turnover ratio which is the ratio of purchases to closingcreditors. Credit Purchases / CreditorsAverage creditors could also be used since an external analyst is unsure whether the yearend numbers are dressed up.If the creditors’ turnover ratio is 5, it means that we paid our outstanding 5 times a year.As long as we do not miss out purchases, the smaller this ratio is, the more efficient is themanagement of creditors.This ratio becomes more understandable if we convert it into number of days. If weturned over creditors 5 times a year, we can say that creditors on an average were 73days. This is called the average days’ purchases in payables and is given by thefollowing formula: 365 / Creditors turnover ratioThe ratio is often called the Average Payment period.D: Profitability RatiosThe profitability ratios measure how efficiently a company manages it assets and howefficiently it manages its operation. The focus is on profits. All of these ratios areexpressed in terms of a percentage.13. Gross profit margin: This is the ratio of gross profit to sales. Gross Profit / SalesThe term gross profit refers to the difference between sales and works cost.Higher the percentage the better it is for the company.Prime Academy FL in CAFM
8. 8. 8 Ratio Analysis14. Operating profit margin: This is the ratio of operating profit to sales. Operating Profit / SalesThe term operating profit is the difference between gross profit and administration andselling overheads. Non operating income and expenses are excluded. Interest expenditureis also excluded because interest is the reward for a particular form of financing and hasnothing to do with operational excellence.Higher the percentage the better it is for the company.15. Net profit margin: This is the ratio of net profit to sales. Net Profit / SalesThe term net profit refers to the final profit of the company. It takes into account allincomes and all expenses including interest costs.Higher the percentage the better it is for the company.16. Return on total assets: This is the ratio of EBIT to Total Assets. EBIT / Total AssetsThe term “total assets” refers to all assets namely net fixed assets and current assets.Higher the percentage the better it is for the company.17. Return on capital employed (ROCE): This is the more popular ratio and is theratio of EBIT to capital employed EBIT / Capital employedThe term “capital employed” refers to the sum of net fixed assets and net working capital.This ratio measures the productivity of money.Higher the percentage the better it is for the company.18. Return on net-worth: This is the ratio of PAT to Net worth. PAT / Net worthThe term “Net-worth” means money belonging to equity share holders and includesreserves net of fictitious assets awaiting write off. It measures how much income a firmgenerates for each rupee stockholders have invested.Higher the percentage the better it is for the company.E: Market RatiosAs these ratios are based on the market price they become crucial numbers to analyse acompany.Prime Academy FL in CAFM
9. 9. Ratio Analysis 919. Earnings per share: This is the ratio of profit after tax and preference dividends tonumber of equity shares outstanding. (Profit after tax – Preference dividend) / No. of equity shares outstandingThis measures the amount of money available per share to equity shareholders.The EPS has to be used with care. Two companies raising identical amounts of moneyand making identical after tax profits can report substantially different EPS.Consider this example. A Ltd. raises Rs.100 lakhs of equity with each share having aface value of Rs.10. The premium on issue is Rs.90 implying that 1,00,000 shares areraised. In accounting speak, Rs.10 lakhs goes to equity account and Rs.90 lakhs goes toshare premium account. Suppose the company makes a profit after tax of Rs.50 lakhs.Since there are 1 lakhs shares outstanding the EPS is Rs.50. The return on net-worth is50%.Now B Ltd. raises Rs.100 lakhs of equity with each share having a face value of Rs.10.The premium on issue is Rs.40 implying that 2,00,000 shares are raised. In accountingspeak, Rs.20 lakhs goes to equity account and Rs.80 lakhs goes to share premiumaccount. Suppose the company makes a profit after tax of Rs.50 lakhs. Since there are 2lakhs shares outstanding the EPS is Rs.25. The return on net-worth is 50%.Both companies have the same RONW, the same face value per share, but the firstcompany returns an EPS of Rs.50 and the second an EPS of Rs.2520. Payout and retention ratio: The payout ratio is the ratio of dividend per share toearnings per share. Dividend per share / EPS Retention ratio is 1 - Payout ratio.21. Price Earnings ratio: This is the ratio of market price per equity share to earningper share. Also known as the PE multiple, the following is the formula: Market price per share / Earnings per share.Suppose the PEM is 12. Typically, this means that if all earnings are distributed asdividends then it would take the investor 12 long years before he recovers his initialinvestment. If that be so, why do investors invest in companies with high PEM? Reason:Investors expect the company’s earnings to grow. The PEM can hence be looked upon asan investor’s confidence in the growth prospects of the company.22. Market to book ratio: This is the ratio of market price per equity share to bookvalue per equity share. The following is the formula: Market price per share / Book value per share.Prime Academy FL in CAFM
10. 10. 10 Ratio AnalysisBook value refers to net-worth. Since book value is an accounting number it reflectshistorical costs. If the value is less than 1 it means that the firm has not been successfuloverall in creating value for the shareholders.Interpreting RatiosWe would like to compare the performance of one company with another (Peer review).If we do that we could immediately run into a problem. For instance, if you wanted tocompare Infosys with Satyam you will have to reckon with the fact that Infosys is by far amuch larger company. It is difficult to even compare Infosys 2002 with Infosys 2007 asthe company’s size would have changed. If you compare Infosys with Microsoft, youhave both a size problem (Infosys is a pigmy compared to Microsoft) and a currencyproblem (Infosys reports in Rs. and Microsoft reports in dollars). The solution lies instandardising the financial statements and this is done by converting all the items fromRs. to percentages. Such statements are called common size statements.Common Size Balance sheet: All items in the Balance sheet are expressed as apercentage of total assets.Common size Income statement: All items in the Profit and Loss account are expressedas a percentage of total sales. This statement tells us what happens to each Rupee ofsales.Trend Analysis: One could fall back on the past. Like, take a look at the ratios acrossthe last five years to understand whether liquidity, solvency, profitability etc. have goneup or come down. This is at the heart of inter-firm comparison.Peer Review: The benchmark could be the industry leader or some company in theindustry which your company wants to catch up with. By comparing your ratios with thebenchmark company, you understand whether you are performing better than thebenchmark company or not.What is most important in the case of ratio analysis is that not all ratios would indicatethings in the same direction. Some would be healthy; others wouldn’t be all that healthy.It takes practice and experience to ascertain trend and interpret. In other words you needto become a good financial doctor. It is hence important that one becomes thorough inthe computation, understanding and interpretation of a few select ratios than in trying tocrack them all. Ratio Analysis is more an art than a science.Limitations1. The RONW is a sacred ratio. But imagine a year when the company decides to write off a major part of its manufacturing facility. Both PAT and Net worth will come down by identical amounts thereby increasing the ratio!2. Then there is the issue of book value. Book value is dangerously susceptible to accounting jugglery and pyro-techniques.Prime Academy FL in CAFM
11. 11. Ratio Analysis 113. There is very little theory to help us identify which ratios to look at and to guide us in establishing benchmarks.4. Very little theory is available to suggest what constitutes a high ratio or a low ratio.5. Different firms use different accounting procedure. Like valuation of inventory.6. Different firms end their fiscal year at different times.7. Trouble with ratios: Different people compute a ratio differently leading to confusion. The specific definitions we use must be spelt out. Those which we are using in this book are the popular usage. When you use ratios to do peer review make sure that the ratios in the two companies are computed in the same way.The DuPont IdentityRatios by themselves mean precious little. If you can understand the link between ratiosand how some ratios can be decomposed to identify the underlying linkages yourappreciation of financial statements and corporate performance will be total. The DuPontCompany used to do just that. We present below a few famous DuPont identities.1. Return on EquityThe Return on Assets or its cousin the Return on Capital Employed talks about theproductivity of money. The Return on Equity is generally higher than the Return onCapital Employed. This is on account of the use of debt financing. For instance, if theROCE is 15%, it means that both debt money and equity money are earning 15%. Now,if debt is rewarded at 8%, it means that the surplus or balance 7% accrues to the equityshareholders. If the debt equity ratio is 1:1 the Return on equity will turn out to be the15% it earns plus the 7% surplus that it pockets from debt namely 22%.Return on Equity is decomposed as under:ROE = PAT/Net-worth = PAT / Net-worth x Assets / Assets = PAT / Assets x Assets / Net-worth = PAT / Assets x Equity MultiplierROE = ROA x (1+Debt-Equity ratio)2. Return on EquityA second decomposition works as under:ROE = PAT / Net-worth = PAT / Net-worth x Assets / Assets = PAT / Assets x Assets / Net-worth = PAT / Assets x Sales / Sales x Assets / Net-worth = Pat / Sales x Sales / Assets x Assets / Net-worthPrime Academy FL in CAFM
12. 12. 12 Ratio AnalysisROE = Profit Margin x TATO x Equity multiplierThe ROE is thus the function of operating efficiency (as measured by profit margin),Asset use efficiency (as measure by total asset turnover) and financial leverage (asmeasured by equity multiplier.ROA, ROE and GrowthIs it possible to know how rapidly a firm can grow! We must remember that over thelong haul, if sales have to grow assets too have to grow because there is only so much thatyou can milk out of an asset. If assets are to grow the firm must find money to fund thesepurchases. The money can come either from internal sources (retention) or externalsources (debt or fresh equity).Internal growth rate: If a company does not want to tap external sources of financingand uses only retained earnings to fund new assets, the rate at which sales can grow isgiven by the following formula: ROA x bInternal growth rate = 1 ROA x bSustainable growth rate (SGR): If a firm relies only on internal financing, over time,the debt equity ratio will decline. Many companies would like to maintain a target debtequity ratio. With this in mind we now lay down the sustainable growth rate on the twinassumptions that (a) company wishes to maintain a target debt-equity ratio and (b) it isunwilling to raise fresh equity. Given these assumptions the maximum growth rate will be ROE x bSustainable growth rate = 1 ROE x bPiecing all these together, we now identify the four drivers of sales growth.1. Profit margin: If the profit margin increases, the internal resources go up. This increases the SGR.2. TATO: An increase in TATO increases the sales per rupee of investment. This decreases the firm’s need for new assets as sales grow and thus increases the sustainable growth rate. If SGR is to3. Financial policy: An increase in the debt equity ratio makes additional debt financing available, thus increasing the SGR. Profit margin4. `Dividend policy: A reduction in dividend payout increases the retention ratio. This TATO increases internally generated funds and thus increases the SGR. Debt Equity Ratio DPPrime Academy FL in CAFM
13. 13. Ratio Analysis 13 Box-1 Categories What they Measure Liquidity ratios Short term solvency Capital Structure Ratio Long term solvency Profitability ratios Ability to make profit Coverage ratios Adequacy of money for payments Turnover ratios Usage of Assets Capital Market ratio Wealth maximisation Box -2 Ratios Formulae Measures Standard RatioI. Liquidity Ratios:1. Current Ratio Current assets The ability of the 1.33 Current Liabilitie s company to use the short term money to repay short term liabilities.2. Quick Ratio Quick assets The ability of the 0.74 Quick Liabilities company to use quick money to repay quick Quick assets liabilities. Current Liabilities3. Cash Reservoir Cash reservoir The readily available cash -Ratio Current Liabilities to meet current liabilities.4. Interval Measure Cash reservoir The no. of days upto - Average daily cash which cash operating expenses can be met with operating expenses available cash reservoir.II. Capital StructureRatios: Prime Academy FL in CAFM
14. 14. 14 Ratio Analysis5. Debt – EquityRatio Debt The financial risk 1.21(i) as ratio Equity involved. High debt-equity ratio is Debt(ii) as percentage risky. Debt Equity6. Capital Gearing Debt Preference -Ratio Equity Debt The financial risk involved. Preference Equity7. Proprietary Ratio Equity Funds High ratio less is the risk. - Net Fixed AssetsIII. ProfitabilityRatios:(a)TurnoverRelated Ratios:8. Gross Profit Ratio Gross Profit Efficiency of the factory. 21% Sales9. Operating Profit Operating Profit Operating efficiency ofRatio the company after taking Sales into account the selling & administration cost.10. Net Profit Ratio Net Profit Overall efficiency of the 4.7% company. Sales(b) InvestmentRelated Ratios11. Return onCapital employed /Return onInvestment (i) Pre – tax EBIT How productively the company utilises its Capital Employed money.Prime Academy FL in CAFM
15. 15. Ratio Analysis 15 (ii) Post – tax PAT Interest How productively the company utilises its Capital Employed money. OR EBIT ( 1 - Tax Rate) Capital Employed12. Return on Equity PAT - Preference dividend How much the 12.7% shareholders earn. Shereholders FundsIV. Coverageratios:13. Interest coverage PAT Interest No. of times earnings are 4.23ratio available to pay interest. Intrest No. of times cash is OR available out of earnings PAT Interest Depreciation to pay interest. Non cash charges Intrest14. Debt - service PAT Interest Depreciation No. of times cash is 1:2coverage ratio available to pay out of Non cash charges OR principle. Principal Interest 1:3V. Turnover Ratios15. Assets Turnover Sales 1.31Ratio Total Assets OR Sales Capital Employed16. Fixed Assets Sales 2.15Turnover Ratio Net Fixed Assets17. Working Capital Sales -Turnover Ratio Working Capital Prime Academy FL in CAFM
16. 16. 16 Ratio Analysis18. Inventory Sales 6.24Turnover Ratio Average Invenory OR Cost of Goods Sold Average Invenory19. Debtors Sales 7.70Turnover Ratio Average Debtors OR Cost of sales Average Debtors20. Creditors PurchasesTurnover Ratio Average CreditorsVI. Velocity Ratios21. Inventory 365 No. of times inventory isVelocity blocked in a year. Inventory Turnover Ratio22. Debtors Velocity 365 How much money are 47.4 days blocked in Debtors. Debtors Turnover Ratio23. Creditors 365 How many days forVelocity which the purchases are Creditors Turnover Ratio outstanding.VII. CapitalMarket Ratios24. EPS PAT - Preference dividend Earning in a year per share. No. of Shares25. PE Multiple Market price No. of times a share is 9.55% EPS being quoted in relation to its earnings.26. Dividend Yield Dividend Dividend received per 14.0% share Market price per share27. Payout Ratio Dividend per share How much paid for every rupee earned. EPSPrime Academy FL in CAFM
17. 17. Ratio Analysis 17 Numerator and Denominator Ratios Formulae Numerator DenominatorI. LiquidityRatios:1. Current Ratio Current assets Inventories + sundry Sundry creditors + Current Liabilitie s debtors + cash + Bank + short term loans + receivables/ accruals + Bank OD+ Cash Prepaid expenses + loans credit + and advances + Outstanding Marketable Investments expenses + Provision for Taxation + Proposed dividends + Unclaimed dividends + other provisions2. Quick Ratio Quick assets Current assets - Current liabilities - Quick Liabilities Inventories - Prepaid Bank OD - Cash expenses credit OR OR OR Quick assets Current assets - Current Liabilities Inventories - Prepaid Current liabilities expenses3. Cash Reservoir Cash reservoir Cash + Bank + Current liabilitiesRatio Current Liabilities Marketable securities + Short term investment OR Current assets - inventories4.Interval Quick assets Current assets - Cost of goods soldMeasure Average daily Inventories - Prepaid + selling, expenses administrative & operating expenses general expenses - depreciation - other non cash expenditures 360 daysII. Capital Prime Academy FL in CAFM
18. 18. 18 Ratio AnalysisStructureRatios:5. Debt - Equity Debt Long term loan + Equity share capitalRatio Equity Short term loan: + Preference share(i) as ratio if it is not payable capital + Reserves Debt within a year even & Surplus - Debt Equity otherwise when the Fictitious assets(ii) as percentage question is silent If it is not protected by securities6. Capital Debt Preference Preference share capital + Equity share capitalGearing Ratio Debentures + Long term + Reserves & Equity loans Surplus - P & L Debt account (Dr. balance) Preference Equity7.Proprietary Proprietary Funds Equity share capital + Fixed Assets +Ratio Preference hare capital + Current assets Total Assets Reserves & Surplus - (excluding Accumulated loss fictitious assets)III. ProfitabilityRatios:(a)TurnoverRelated Ratios:8. Gross Profit Gross Profit Gross profit as per Sales net of returnsRatio (as %) x 100 Trading Account Sales9. Operating OperatingProfit Gross profit - Non- Sales net of returnsProfit Ratio (as x 100 opearting expenses + Sales%) Non-opearating income10. Net Profit Net Profit Net profit as per Profit & Sales net of returnsRatio x 100 Loss account Sales(as %)(b) InvestmentRelated Ratios11. Return onCapital employed/ Return onInvestmentPrime Academy FL in CAFM
19. 19. Ratio Analysis 19 (i) Pre- tax EBIT Net Profit after Tax + Equity Share Tax + Interest + Non - Capital + Capital Employed trading Expenses + Non - Preference Share operating Incomes. Capital + Reserves & Surplus + Debentures - Loss - Non-trading investment. (ii) Post - tax PAT Interest Profit after Tax + Interest Equity Share Capital + Capital Employed Preference Share OR Capital + Reserves & Surplus + EBIT ( 1 - Tax Rate) Debentures - Loss Capital Employed - Non-trading investment.- Preliminary expenses12. Return on PAT - Preference Profit after Tax - Equity ShareEquity Preference dividend Capital + dividend Preference Share (Equity earnings) Shereholders Funds Capital + Reserves & Surplus - LossIV. Coverageratios:13. Interest PAT Interest Net Profit after Tax + Interest on Loancoverage ratio Tax + Interest + Non - (Long term & short Intrest trading Expenses + Non - tem) OR operating Incomes. PAT Interest Depreciation Non cash charges Intrest14. Debt - service PAT Interest Net profit as per P & L Interest on debt +coverage ratio account - Tax + Interest + installment of debt Depreciati on Non - trading Expenses + Non cash charges Non - operating Incomes. Principal InterestV. TurnoverRatios Prime Academy FL in CAFM
20. 20. 20 Ratio Analysis15. Assets Sales Sales net of return Net fixed Assets +Turnover Ratio Current assets Total Assets (excluding OR fictitious assets) Sales Capital Employed16. Fixed Assets Sales Sales net of return Net fixed AssetsTurnover Ratio Net Fixed Assets (Fixed assets - Depreciation)17. Working Sales Sales net of return Current assets -Capital Turnover current liabilities Working CapitalRatio18. Inventory Sales Opening stock +Turnover Ratio Closing stock Average Invenory Sales net of return OR 2 OR Cost of production - Cost of Goods Sold Closing stock of finished Average Invenory goods19. Debtors Sales Net credit sales Opening debtors +Turnover Ratio Closing debtors Average Debtors OR Cost of goods sold + 2 OR Administration exp. + Cost of sales Selling & Distribution Average Debtors exp.20. Creditors Purchases Net credit purchases Opening creditors +Turnover Ratio Closing creditors Average Creditors 2VI. VelocityRatios21. Inventory 365Velocity Inventory Turnover RatioPrime Academy FL in CAFM
21. 21. Ratio Analysis 2122. Debtors 365Velocity Debtors Turnover Ratio23. Creditors 365 .Velocity Creditors Turnover RatioVII. CapitalMarket Ratios24. EPS PAT - Preference PAT - Preference No. of equity shares dividend dividend No. of Shares25. PE Multiple Market price Current market price of EPS EPS equity share26. Dividend Dividend Dividend Current marketYield price of equity Market price per share share27. Payout Ratio Dividend per share Dividend per share EPS EPSRatio Analysis compares one financial figure with another. The current ratio isaffected by various types of transactions. For example suppose the form borrows Prime Academy FL in CAFM
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https://www.numere-romane.ro/cum_se_scrie_numarul_arab_cu_numerale_romane.php?nr_arab=110487&nr_roman=(C)(X)CDLXXXVII&lang=en
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# Convert number: 110,487 in Roman numerals, how to write?
## Latest conversions of Arabic numbers to Roman numerals
110,487 = (C)(X)CDLXXXVII Nov 30 04:56 UTC (GMT) 2,016,194 = (M)(M)(X)(V)MCXCIV Nov 30 04:56 UTC (GMT) 19,446 = (X)M(X)CDXLVI Nov 30 04:56 UTC (GMT) 56 = LVI Nov 30 04:56 UTC (GMT) 111,904 = (C)(X)MCMIV Nov 30 04:56 UTC (GMT) 19,447 = (X)M(X)CDXLVII Nov 30 04:56 UTC (GMT) 30,005 = (X)(X)(X)V Nov 30 04:56 UTC (GMT) 1,989 = MCMLXXXIX Nov 30 04:56 UTC (GMT) 221,282 = (C)(C)(X)(X)MCCLXXXII Nov 30 04:56 UTC (GMT) 1,203,821 = (M)(C)(C)MMMDCCCXXI Nov 30 04:56 UTC (GMT) 1,945 = MCMXLV Nov 30 04:56 UTC (GMT) 16,092 = (X)(V)MXCII Nov 30 04:56 UTC (GMT) 154,351 = (C)(L)M(V)CCCLI Nov 30 04:56 UTC (GMT) converted numbers, see more...
## The set of basic symbols of the Roman system of writing numerals
• ### (*) M = 1,000,000 or |M| = 1,000,000 (one million); see below why we prefer this notation: (M) = 1,000,000.
(*) These numbers were written with an overline (a bar above) or between two vertical lines. Instead, we prefer to write these larger numerals between brackets, ie: "(" and ")", because:
• 1) when compared to the overline - it is easier for the computer users to add brackets around a letter than to add the overline to it and
• 2) when compared to the vertical lines - it avoids any possible confusion between the vertical line "|" and the Roman numeral "I" (1).
(*) An overline (a bar over the symbol), two vertical lines or two brackets around the symbol indicate "1,000 times". See below...
Logic of the numerals written between brackets, ie: (L) = 50,000; the rule is that the initial numeral, in our case, L, was multiplied by 1,000: L = 50 => (L) = 50 × 1,000 = 50,000. Simple.
(*) At the beginning Romans did not use numbers larger than 3,999; as a result they had no symbols in their system for these larger numbers, they were added on later and for them various different notations were used, not necessarily the ones we've just seen above.
Thus, initially, the largest number that could be written using Roman numerals was:
• MMMCMXCIX = 3,999.
| 700 | 2,095 |
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# math
posted by .
what is the derivative of 2^x?
I know that it's something with ln but I can't find it in my notes
• math -
d/dx (a^x) = a^x *ln a
• math -
thanks! i just found it in my notes too
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In a purse there are 20-rupee notes 10-rupee notes and 5-rupee notes. the number of 5-rupee notes exceeds two times the 10-rupee notes by one. the 20-rupee notes are 5 less than the 10-rupee notes. if the total value of the money in …
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http://www.maplesoft.com/support/help/Maple/view.aspx?path=Task/BuildStructuredListOrderedPairs
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Build a Structured List of Ordered Pairs - Maple Help
Build a Structured List of Ordered Pairs
Description Build a structured list of ordered pairs from two lists of the same length.
Enter the first list.
> $\mathrm{L1}:=\left[{-}{5}{,}{-3}{,}{-2}{,}{0}{,}{1}{,}{3}{,}{4}{,}{6}\right]$
${\mathrm{L1}}{:=}\left[{-}{5}{,}{-}{3}{,}{-}{2}{,}{0}{,}{1}{,}{3}{,}{4}{,}{6}\right]$ (1)
Enter the second list.
> $\mathrm{L2}:=\left[{-}{4}{,}{3}{,}{2.7}{,}{0}{,}{-2}{,}{-5.9}{,}{0}{,}{10}\right]$
${\mathrm{L2}}{:=}\left[{-}{4}{,}{3}{,}{2.7}{,}{0}{,}{-}{2}{,}{-}{5.9}{,}{0}{,}{10}\right]$ (2)
Build a structured list of ordered pairs using the 2 defined lists.
>
$\left[\left[{-}{5}{,}{-}{4}\right]{,}\left[{-}{3}{,}{3}\right]{,}\left[{-}{2}{,}{2.7}\right]{,}\left[{0}{,}{0}\right]{,}\left[{1}{,}{-}{2}\right]{,}\left[{3}{,}{-}{5.9}\right]{,}\left[{4}{,}{0}\right]{,}\left[{6}{,}{10}\right]\right]$ (3)
>
Commands Used
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# flvs
posted by .
Your teacher has a jar of chocolate candies. For every completed assessment, she gives herself three pieces of chocolate.
candies eaten after 67 completed assessments
• flvs -
3 * 67 = ?
• flvs -
Use calculator do solve. 67 times 3 is 201.
| 68 | 265 |
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https://www.physicsforums.com/threads/cos-x-question-how-would-you-call-x.528717/
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Cos(x) question! How would you call 'x'?
1. Sep 10, 2011
mimzy
So I have this integration to solve but I needed help and I just couldn't make up a term to call that 'x' ><!! the term that is inside the parenthesis!! I know I've learned it somewhere but I just can't remember and it's making me nuts! >A<
Also, what would you do in order to solve an equation that involves lets say sin(x)/cos(2x) [its just made up so I can explain myself a little better :D] and you need to combine both terms... but in order to combine them you gotta make a substitution for that 'x' term ><
2. Sep 10, 2011
Caramon
Integration by Substitution? Let x = $$\theta$$? The symbol $$\theta$$ is pronounced theta? No idea what you are talking about.
Just so you know, there is no simpler form to:
$$\frac{sin(\theta)}{cos(2 \theta)}$$
IT DOES NOT EQUAL:
$$tan(\frac{1}{2} \theta)$$
No idea if this helps, still have no clue what you are asking...
3. Sep 10, 2011
mimzy
im just wondering for the name of that theta since it wont always be the same thing on all equations... it changes depending on what u are asked for... like x, theta, pi and so on... just he generic name of that >< sorry if I wasn't clear enough
4. Sep 10, 2011
Caramon
A variable? There's no set word to describe whatever variable you insert inside of a trigonometric function. You just say that it is the parameter in terms of which the function is defined...?
5. Sep 10, 2011
Hurkyl
Staff Emeritus
This is a function application expression:
$$A(B)$$
It is composed of two subexpressions:
• The expression $A$, which should be of function type
• The expression $B$, whose type should be contained in domain of $A$
In such expressions, $B$ is sometimes called the "argument", such as in the sentence "$B$ is the argument passed into the function $A$".
6. Sep 10, 2011
Nowhere Man
Speaking as a programmer, given cos(x), x is the argument or parameter to the function cos(). It can be a literal, such as 20, a variable, or an expression.
Fred
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First, to define the functions themselves. 1, Functions of two variables p. The simplest method to swap two variables is to use a third temporary variable :. experiment to a function X(t,e). Very easy to understand!Prealgebra exponent lessons, examples and practice problems Algebra Lessons at Cool math. Derivatives told us about the shape of the function, and let us find local max and min - we want to be able to do the same thing with a function of two variables. A function of a single input variable observations has been created from the two-input variable function fitdistr: fixing one of the input variables by setting densfun = "normal". I found a and b for several values of x2, so I do have equations f(x1) for some fixed x2. In single-variable calculus, you learned how to compute the derivative of a function of one variable, y= f(x), with respect to its independent variable x, denoted by dy=dx. In a two-variable problem rewrite the equations so that when the equations are added, one of the variables is eliminated, and then solve for the remaining variable. The multiple integral is a definite integral of a function of more than one real variable, for example, f(x, y) or f(x, y, z). You can use fminsearch to optimize your coefficients, but you still need to know the basic form of the function. Although the formal proof is not trivial, the variable-dependence diagram shown here provides a simple way to remember this Chain Rule. There is another way-a highly engaging way that does not neglect readers' own intuition, experience, and excitement. Average value of a function To find the average value of a function of two variables, let's start by looking at the average value of a function of one variable. This happens when you get a “plus or minus” case in the end. Create a function of two variables. One-variable calculus makes extensive use of graphs in or-. Now the UNION of A and B, written A B = (1,2,3,4,5). In mathematics, the result of a modulo operation is the remainder of an arithmetic division. There are three problems, each of which has a background discussion, an illustrative example, and an exercise for you to do. Functions f (x1, x2, , xn) of n variables, Symmetry. The Effective Use of Graphs. Modern code has few or no globals. For a thermal contact between the two put a thermal conductance value. In particular, a function of 2 variables is a function whose inputs are points ( x , y ) in the xy -plane and whose outputs real numbers. The inputs are ordered pairs, (x, y). User asks to enter the value. The outputs are real numbers. The Cumulative Distribution Function for a Random Variable \ Each continuous random variable has an associated \ probability density function (pdf) 0ÐBÑ \. The area of the triangle and the base of the cylinder: A= 1 2 bh. We also write z = f (x ,y ) The variables x and y are independent variables and z is the. Use the Show menu to switch from one mode to another. The value of num1 and num2 are initialized to variables a and b respectively. functions of several variables and partial differentiation (2) The simplest paths to try when you suspect a limit does not exist are below. The add-on store offers several custom functions as add-ons for Google Sheets. Usually this follows easily from the fact that closely related functions of one variable are continuous. as subroutines, routines, procedures, methods, or subprograms. Definition 1. If you define global variables (variables defined outside of any function definition), they are visible inside all of your functions. A swapping function: To understand how explicit pass by reference of parameters works in C, consider implementing a swap function in C, that is, a function that passes in two variables and swaps their values. Definition 1. 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Part A: Functions of Two Variables, Tangent Approximation and Opt; Part B: Chain Rule, Gradient and Directional Derivatives; Part C: Lagrange Multipliers and Constrained Differentials; Exam 2. Solve this system of equations by using substitution. You define a function in much the same way you define a variable. Use the debugger to see what's the mismatch in dimensions; it's not totally apparent as one would presume i is a loop index and so is a single integer value; if MS3 is an array it would also be a single value but if it happened to be a function it could return something other than. Functions of Several Variables This manual contains solutions to odd-numbered exercises from the book Functions of Several Vari-ables by Miroslav Lovri´c, published by Nelson Publishing. Note that it is assumed that the two lists given in the table command are both factors. Economists of this period, while recognizing that the law of diminishing returns (or the law of variable proportions) applied when units of a variable. However, there is also a main di⁄erence. Find the standard deviation of the eruption duration in the data set faithful. Identities Proving Identities Trig Equations Trig Inequalities Evaluate Functions Simplify Statistics Arithmetic Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge. The added risk brought on by the complexity of machine-learning models can be mitigated by making well-targeted modifications to existing validation frameworks. For functions of two or three variables the situation is more complicated because there are infinitely. • Matlab has several different functions (built-ins) for the numerical. Staffing: After a manager discerns his area's needs, he may decide to beef up his staffing by recruiting, selecting, training, and developing employees. We now extend this concept to functions of two variables. com - Functions. Swapping two variables refers to mutually exchanging the values of the variables. When we considered functions and graphs of one variable, one of the first things we did was to transform those graphs through shifts and stretches. De nition A critical point (x0;y0) of fis a point where both the partial derivatives @f=@xand @f=@y. If you will need guidance with algebra and in particular with ordered pairs and inequalies online calculator or fractions come visit us at Algebra-equation. For a function of one variable, a function w = f (x) is differentiable if it is can be locally approximated by a linear. Active 2 years, 7 months ago. 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These are special variables that take on the values that you give when you call for the function, meaning you can give it any two numbers and it can add them together. Because the correlation between reading and mathematics can be determined in the top section of the table, the correlations between those two variables is not repeated in the bottom half of the table. of Manchester) 5 2 Functions of multiple [two] variables In many applications in science and engineering, a function of interest depends on multiple. The purpose of this lab is to give you experience in applying calculus techniques relating to finding extrema of functions of two variables. Algebra functions lessons with lots of worked examples and practice problems. Alternatively, the function also knows it must return the first argument, if the value of the "number" parameter, passed into the function, is equal to "first". Definition of Mathematical Expectation Functions of Random Variables Some Theorems on Expectation The Variance and Standard Deviation Some Theorems on Variance Stan-dardized Random Variables Moments Moment Generating Functions Some Theorems on Moment Generating Functions Characteristic Functions Variance for Joint Distribu-tions. To use or explore these add-ons: Create or open a spreadsheet in Google Sheets. The partial derivative of f with respect to y can similarly be found by treating x as a constant whenever it appears. second variable y appears, it is treated as a constant in every respect. The concept of the graph of a function is generalized to the graph of a relation. Part 1: Functions of 2 Variables. input variables and other variables you create within the function and in doing so, you create the output variables you desire. You have now created a function called sum. One Function of Two Random Variables Given two random variables X and Y and a function g(x,y), we form a new random variable Z as Given the joint p. This will help us to see some of the interconnections between what can seem like a huge body of loosely related de nitions and theorems1. Functions 3D Plotter is an application to drawing functions of several variables and surface in the space R3 and to calculate indefinite integrals or definite integrals. (a) True, and I am very con dent (b) True, but I am not very con dent (c) False, but I am not very con dent (d) False, and I am very con dent 2. Example 3: Using the function from Example 2, describe and graph the following functions: (i) f(x, y) = 3 - x2 - y2. Note that it is assumed that the two lists given in the table command are both factors. For in-stance, step functions are continuous except at their steps, that is, where there are jump discontinu-ities. In the next two sections we introduce these two concepts and develop some of their properties. 10 Two-Dimensional Random Variables Definition 1. Addition of two numbers in C For example, if a user will input two numbers as; '5', '6' then '11' (5 + 6) will be printed on the screen. Let (X;d)and (Y;d′)be two metric spaces, A ⊆X a nonempty set, a function f ∶A →Y and x. f Obviously. Derivatives told us about the shape of the function, and let us find local max and min - we want to be able to do the same thing with a function of two variables. Relation with other tests Changing the number of variables. We also write z = f (x ,y ) The variables x and y are independent variables and z is the. even functions of one variable may have both maximum and minimum points). The function writePictureTo takes two parameters: the picture variable and the pathname. First-order partial derivatives of functions with two variables. The area of a circle is a function of -- it depends on -- the radius. 1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. In the expression (c = a + b) overflow may occur if the sum of a and b is larger than the maximum value which can be stored in the variable c. In the short run, production function is explained with one variable factor and other factors of productions are held constant. ) Variables and functions in all parts of a makefile are expanded when read, except for the shell commands in rules, the right-hand sides of variable definitions using `=', and the bodies of variable definitions using the define directive. Now you know the basics of using two variable -- or complex -- functions. These are special variables that take on the values that you give when you call for the function, meaning you can give it any two numbers and it can add them together. We recall that a critical point of a function of several variables is a point at which the gradient of the function is either the zero vector 0 or is undefined. Let f : D ⊂ R → R and let a ∈ R. To plot the point (2,3), for example, you start at the origin Independent and Dependent Variables. Scalar functions of two variables Our main goal in this tutorial is to explore ways to plot functions of two variables. f how does one obtain. 2 Graphs should always have at minimum a caption, axes and scales, symbols, and a data field. De nition A critical point (x0;y0) of fis a point where both the partial derivatives @f=@xand @f=@y. Usually, there is more than one correct answer. And the fun part with these guys is that you can just kind of, imagine a fluid flowing, so here's a bunch of droplets, like water, and they kind of flow along that. x^2*y+x*y^2 ） The reserved functions are located in " Function List ". I am trying to create the interpolating function for a function of two variables, over a finite area. It can be used as a worksheet function (WS) in Excel. • Matlab has several different functions (built-ins) for the numerical. For a function of a single variable there are two one-sided limits at a point x0, namely, lim x!x+ 0 f(x) and lim x!x 0 f(x) reflecting the fact that there are only two directions from which x can approach x0, the right or the left. Concave functions of two variables While we will not provide a proof here, the following three definitions are equivalent if the function f is differentiable. There is no need to list the 3 twice. Definition of Variables and Examples. identically distributed Exponential random variables with a constant mean or a constant parameter (where is the rate parameter), the probability density function (pdf) of the sum of the random variables results into a Gamma distribution with parameters n and. accept a wide variety of mathematical expressions. To close the answer window and get back to the quiz, click on the X in the upper right corner of the answer window. Equations of a Straight Line. Functions of two variables 1. AMS 311 Joe Mitchell Examples: Joint Densities and Joint Mass Functions Example 1: X and Y are jointly continuous with joint pdf f(x,y) = ˆ cx2 + xy 3 if 0 ≤ x ≤ 1, 0 ≤ y ≤ 2. Functions can be recognized, described, and examined in a variety of ways, including graphs, tables, and sets of ordered pairs. Could someone please explain a function of two variables to me. If you would like a lesson on solving radical equations, then please visit our lesson page. Hence the square of a Rayleigh random variable produces an exponential random variable. Therefore, in order to be able to. The applet initially starts in the Input mode, which lets you choose a function to plot (you can either enter it manually, or select one from the drop-down list; click on the Plot button to create the new plot). Observe that because of the non-negativity constraint, the sum of any collection of variables cannot be negative. Functions of Two Variables. Variable b1 and b2 are baseline variables. Laval (KSU) Functions of Several Variables Today 14 / 22. peaks is a function of two variables, obtained by translating and scaling Gaussian distributions, which is useful for demonstrating mesh, surf, pcolor, contour, and so on. One important similarity to notice between the limit of a one variable function and the limit of a two variable function is that $\sqrt{(x - a)^2 + (y - b)^2}$ represents the distance between the point $(x, y)$ and $(a, b)$ in $\mathbb{R}^2$. First, we will create an intensity image of the function and, second, we will use the 3D plotting capabilities of. My function is exponential for x1 with two coefficients that depend on x2: f(x1,x2)=a*(x1)^b, where a and b are functions of x2. Files are available under licenses specified on their description page. Under the pass-by-value mechanism, the parameter variables within a function receive a copy of the variables ( data ) passed to them. The scatter plot plots the points (x, y) where x is a value from one data list (Xlist) and y is the corresponding value from the other data list (Ylist). A function f(x, y) of two independent variables has a maximum at a point (x 0, y 0) if f(x 0, y 0) f(x, y) for all points (x, y) in the neighborhood of (x 0. Laval (KSU) Functions of Several Variables Today 14 / 22. The outputs are real numbers. As the n -tuple x = (x1, x2, , xn) varies in X, a subset of ℝn, Implicit functions. Chain Rule And Composite Functions Derivative of Composite Function with the help of chain rule: When two functions are combined in such a way that the output of one function becomes the input to another function then this is referred to as composite function. Lady (September 5, 1998) There are three ways that a function can be discontinuous at a point. Distributions of Functions of Random Variables 1 Functions of One Random Variable Case of two-to-one transformations. You can choose any other combination of numbers as well. First, we will create an intensity image of the function and, second, we will use the 3D plotting capabilities of matplotlib to create a shaded surface plot. 4 Higher partial derivatives Notice that @f @x and @f @y are themselves functions of two variables, so they can also be partially differenti-ated. Functions of three variables are similar in many aspects to those of two variables. Because we're trying to keep things a little bit simpler, we'll concentrate on functions of two variables. Limits of a Rational Function of Two Variables Roger B. When variables change together, their interaction is called a relation. The dependent variable is what is affected by the independent variable-- your effects or outcomes. In the above example, two variables, num1 and num2 are passed to function during function call. Example 3: Using the function from Example 2, describe and graph the following functions: (i) f(x, y) = 3 - x2 - y2. Topic 5: Functions of multivariate random variables † Functions of several random variables Sum of 2 random variables † Let X and Y be two random variables. These are just constant functions, and because of that, degree 0 polynomials are often called constant polynomials. Equations of a Straight Line. For functions of two or three variables the situation is more complicated because there are infinitely. This firm minimizes its cost of producing any given output y if it chooses the pair (z 1, z 2) of inputs to solve the problem. Average value of a function To find the average value of a function of two variables, let's start by looking at the average value of a function of one variable. It seems reasonable, and can be shown to be true,. FUNCTION OF TWO VARIABLES Definition: A variable Z is said to be a function of two independent variables x and y denoted by z=f (x,y) if to each pair of values of x and y over some domain D f ={(x,y): a type:. You have now created a function called sum. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. which is the density for an exponential random variable with parameter = 1/(2 2a), as can be seen from inspection of (2-27). Sometimes it will be preferable to think of f as taking one (2-dimensional) vector input instead of two scalar inputs. Optimization Problems with Functions of Two Variables. Our discussion is not limited to functions of two variables, that is, our results extend to functions of three or more variables. As with single variable functions, two classes of common functions are particularly useful and easy to describe. The variables held fixed are viewed as parameters. User make a function named swap that will be called in other class. For a function of n variables it can be a maximum point, a minimum point or a point that is analogous to an inflection or saddle point. two variables y et z is put equal to zero, then either variable is defined by the other and thus a function of this variable emerges, since before they were not mutually dependent. Write a script m-flle and use the Control System Toolbox functions ss and ltiview to form the state model and its step response. Furthermore, sums, dif-. of Manchester) 5 2 Functions of multiple [two] variables In many applications in science and engineering, a function of interest depends on multiple. • Matlab has several different functions (built-ins) for the numerical. The purpose of this lab is to give you experience in applying calculus techniques relating to finding extrema of functions of two variables. I suspect I will need the surface chart but can some one tell me how to generate the chart and what to enter on the worksheet. Under the pass-by-value mechanism, the parameter variables within a function receive a copy of the variables ( data ) passed to them. †Forcontinuous randomvariables. The set D is the domain of f and its range is the set of values that f takes on. Following are different ways. Does anyone know of any helpful tutorials that will help me get the Domain and range, functions of 2 variables | Physics Forums. The add-on store offers several custom functions as add-ons for Google Sheets. For a thermal contact between the two put a thermal conductance value. In the case of functions of two variables, that is functions whose domain consists of pairs (x, y), the graph can be identified with the set of all ordered triples ((x, y, f(x, y)). Two expressions involving template parameters are called equivalent if two function definitions that contain these expressions would be the same under ODR rules, that is, the two expressions contain the same sequence of tokens whose names are resolved to same entities via name lookup, except template parameters may be differently named. I will give the definition of differentiablity in 2D. Quotient of two random variables. For a function of one variable, a function w = f (x) is differentiable if it is can be locally approximated by a linear. There is no need to list the 3 twice. The sum of two incomes, for example, or the difference between demand and capacity. 2 Limits and Continuity of Functions of Two Variables In this section, we present a formal discussion of the concept of continuity of functions of two variables. More generally, if two or three variables are changing, how do we explore the correspondingchangein w? The answer to these questionsstarts with the generalizationof the idea of the differential as linear approximation. Local extreme values of a function of two variables. Limits and Continuity of Functions of Two or More Variables Introduction. 1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. The euclidean_division function to calculate online the quotient and the remainder in the euclidean division of two polynomials or two integers. This gives a nice graphical representation where the plane at x = 0 bounds the function from below. Continuous Random Variables Acontinuous random variable X takes values in an interval of real numbers. functions of two variables. In this paper distribution of zeros of solutions of functional equations in the space of functions of two variables is studied. So far, we have discussed how we can find the distribution of a function of a continuous random variable starting from finding the CDF. Functions f (x1, x2, , xn) of n variables, Symmetry. That is, a function expresses dependence of one variable on one or more other variables. Let us assume that both f and as many partial derivatives as necessary are continuous near (x 0,y 0). Recall that the definition of the limit of such functions is as follows. More information about applet. In single-variable calculus we were concerned with functions that map the real numbers $\R$ to $\R$, sometimes called "real functions of one variable'', meaning the "input'' is a single real number and the "output'' is likewise a single real number. Not only for computing the variance of the transformed variable Y, but also for its mean. When variables change together, their interaction is called a relation. 1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. For a function of a single variable there are two one-sided limits at a point x0, namely, lim x!x+ 0 f(x) and lim x!x 0 f(x) reflecting the fact that there are only two directions from which x can approach x0, the right or the left. The graph of is a subset of three-dimensional Euclidean space with coordinates , given by the equation: Equivalently, it is the set of points: Pictorially, this graph looks like a surface for a nice enough function. $\endgroup$ - Gerhard Paseman Feb 13 at 18:10. Fortunately, the functions we will examine will typically be continuous almost everywhere. The function makes it possible to verify by using the Pythagorean theorem knowing the lengths of the sides of a triangle that this is a right triangle. For a continuous real-valued function of two real variables, the graph is a surface. Most useful functions of one variable are con-tinuous, but there are a few exceptions. characterizations, namely, the mass function for discrete random variable and the density function for continuous random variables. 3-Dimensional graphs of functions are shown to confirm the existence of these points. In a "system of equations," you are asked to solve two or more equations at the same time. Re: st: computing covariance. com, a free online graphing calculator. To evaluate z, first create a set of (x,y) points over the domain of the function using meshgrid. functions of several variables and partial differentiation (2) The simplest paths to try when you suspect a limit does not exist are below. 1 Visualizing functions of 2 variables One problem with thinking about functions of several variables is that they can be harder to picture than functions of just one variable. Imagine a surface, the graph of a function of two variables. Functions of 2 Variables Functions and Graphs In the last chapter, we extended di⁄erential calculus to vector-valued functions. AMS 311 Joe Mitchell Examples: Joint Densities and Joint Mass Functions Example 1: X and Y are jointly continuous with joint pdf f(x,y) = ˆ cx2 + xy 3 if 0 ≤ x ≤ 1, 0 ≤ y ≤ 2. Hence, time is always on the X axis. Functions 3D Plotter and Analytic double integrator Functions 3D Plotter is an on line app to plotting two-variabled real functions, ie functions of type f(x,y) or with more precision f: R 2 → R (x,y) → f(x,y) 3D Functions Plotter calculates double integrals in analytic or numeric form. If the relation is not a function the graph contains at least two points with the same x-coordinate but with different y-coordinates. There is a probability associated with X falling between two numbers a weekday ) ) ) { $datemonth =$wp_locale->get_month( $datefunc( 'm',$i ) ); $datemonth_abbrev =$wp_locale->get_month_abbrev. For functions of two or three variables the situation is more complicated because there are infinitely. Free online 3D grapher from GeoGebra: graph 3D functions, plot surfaces, construct solids and much more!. Again, please enter this line into. Functions of more variables can be defined similarly. It would be useful to read these two guides. One primary difference, however, is that the graphs of functions of more than two variables cannot be visualized directly, since they have dimension greater than three. The concept of the graph of a function is generalized to the graph of a relation. Thread: chart function of two variables. Furthermore, sums, dif-. For example, if you are studying the effects of a new educational program on student achievement, the program is the independent variable and your measures of achievement are the dependent ones. You can choose any other combination of numbers as well. f(x,y) is inputed as "expression". Fortunately for us, we have technology which facilitates this task. Use Wolfram|Alpha to generate plots of functions, equations and inequalities in one, two and three dimensions. But polynomials, trig functions, power and root functions, logarithms, and exponential func-tions are all continuous. Suppose that X and Y are two random variables having moment generating functions MX(t) and MY (t) that exist for all t in some interval 3. In the present case, we see that the critical point at the origin is a local maximum of f2 , and the second critical point is a saddle point. Integrals of a function of two variables over a region in R 2 are called double integrals, and integrals of a function of three variables over a region of R 3 are called triple integrals. The standard deviation of an observation variable is the square root of its variance. It is good programming practice to avoid defining global variables and instead to put your variables inside functions and explicitly pass them as parameters where needed. I'm having a bit of trouble grasping the domain and range of functions of 2 variables. 3-Dimensional graphs of functions are shown to confirm the existence of these points. Boolean Functions (Expressions) It is useful to know how many different Boolean functions can be constructed on a set of Boolean variables. To input the variable x as a Wildcard, first type Shift + ?, then type x; similarly, for y. Importantly,. peaks is a function of two variables, obtained by translating and scaling Gaussian distributions, which is useful for demonstrating mesh, surf, pcolor, contour, and so on. Local extreme values of a function of two variables. Dependent has two categories, there is only one discriminant function. When variables change together, their interaction is called a relation. In general, I can't create new functions in a poisoned session. That is, a function expresses dependence of one variable on one or more other variables. Hi, It is possible to define a function of two variables using an interpolation function defined in a text file using the spreadsheet data format: the first column contains the values for the first input argument, the second column contains the values for the seond input argument, and the third column contains the function's value. Function definition, the kind of action or activity proper to a person, thing, or institution; the purpose for which something is designed or exists; role. So this is more like a re-visit to the good old topic. As the other answer shows, the mere existence of partial derivatives doesn't even guarantee that the function is continuous. Notice we kept that one dimensional distance in our limit definition for functions of two variables when we said |f(x, y) - L| < e. Applications of Extrema of Functions of Two Variables. Although the formal proof is not trivial, the variable-dependence diagram shown here provides a simple way to remember this Chain Rule. The multiple integral is a definite integral of a function of more than one real variable, for example, f(x, y) or f(x, y, z). If not, then we will want to test some paths along some curves to first see if the limit does not exist. Also, use ss2tf to obtain the fllter’s transfer function. Fortunately for us, we have technology which facilitates this task. You can create a two way table of occurrences using the table command and the two columns in the data frame: In this example, there are 51 people who are current smokers and are in the high SES. \+,œTÐ+Ÿ\Ÿ,Ñœ0ÐBÑ. 16 Possible Functions of Two Variables. ” For example, how much you weigh is related (correlated) to how much you eat. Correlation look at trends shared between two variables, and regression look at causal relation between a predictor (independent variable) and a response (dependent) variable. f Obviously. In case of two independent variables X 1 and X 2 such a function may be expressed as under: Y = a + bX 1 - cX 2 1 + dX 2 - eX 2 2. First, we introduce the de nition of a function of two variables: A scalar-valued. Equations of a Straight Line. The standard deviation of an observation variable is the square root of its variance. Polynomial Calculator. In mathematics, the result of a modulo operation is the remainder of an arithmetic division. Graph the function f(x,y) = xy using x,y,z-coordinate axes in 3-D space. It seems reasonable, and can be shown to be true,. Just for consistency we can think of a function:. Loading Graph Functions of 2 Variables. lang package, and not in the java. In the case of functions of two variables, that is functions whose domain consists of pairs (x, y), the graph can be identified with the set of all ordered triples ((x, y, f(x, y)). 2 to find the resulting PDFs. When we extend this notion to functions of two variables (or more), we will see that there are many similarities. For functions of two or three variables the situation is more complicated because there are infinitely many. I am now trying to find a general equation f(x1,x2). Long weekends and highway traffic on Friday afternoon C. First-order partial derivatives of functions with two variables. Applications of Extrema of Functions of Two Variables. Could someone please explain a function of two variables to me. function of two variables is far more di¢ cult than a function of one variable. Calculates the table of the specified function with two variables specified as variable data table. The INTERSECTION of two sets is the set of elements which are in both sets. You define a function in much the same way you define a variable. The value of num1 and num2 are initialized to variables a and b respectively. Function composition. The partial derivative of a function of two or more variables with respect to one of its variables is the ordinary derivative of the function with respect to that variable, considering the other variables as constants. In the example above, the diagonal was used to report the correlation of the four factors with a different variable. time) and one or more derivatives with respect to that independent variable. y(s;t) and z(s;t), are called the component functions of the vector-valued function g. The Method of Transformations. Examples 4. Here that means you need to use the. I found a and b for several values of x2, so I do have equations f(x1) for some fixed x2. Furthermore, sums, dif-.
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The Difference Between Mass and Weight
Mass and weight are two measurements that often get confused. The amount of matter that something has is its mass, but its weight changes depending on the gravitational force acting on it.
The most common way to measure mass is with a balance. Let’s explore some other methods of determining an object’s mass.
What is Mass?
Most students are taught to weigh things using a balance. That is a great way to introduce the concepts of weight and mass, but it doesn’t teach the real definition of each term.
Mass is a property of matter, regardless of its location in the universe. It is a fundamental quantity with the SI unit of kilogram (kg).
Weight is a force that depends on gravitational attraction. Two objects of the same size can have different weights because gravity affects them differently. An object’s weight can change, for example, when it is moved to a different planet with a stronger or weaker gravity. However, the object’s mass will stay the same. Many people get the two terms confused and use them interchangeably, but they are different measurements.
Gravitational Force
The gravitational force that exists between objects with mass attracts them and causes them to fall toward each other. This is a universal law of nature that was first postulated by Sir Isaac Newton in 1687. Gravity is inversely proportional to the square of the distance between the centers of the two masses and increases with the mass of the objects.
In technical contexts, engineers use the term kilogram-force to describe the standard value of gravity (symbol: G) at Earth’s surface—9.80665 m/s2—and they convert mass to a corresponding unit of force in newtons. Objects weigh differently on different planets, depending on their size and the strength of their gravity, but they always have the same mass.
Ever since the 17th century, scientists have tried to measure the strength of gravity in a laboratory. The most precise technique uses a torsion balance. Students can experiment with this equipment and record the results on graph paper. They can also write an equation from the data to show the relationship between the force of gravity and the mass of the object.
Weighing Scales
A scale is the instrument used to measure the amount of matter in an object. It can be used in a variety of applications, from measuring a person’s body weight to weighing ingredients for baking. Weighing scales are also commonly found in chemistry labs and other scientific settings.
The most common scales used in mass measurement are balances, which compare unknown masses to a known quantity – in this case standard weights. This allows the scale to provide a reading that is independent of changes in gravity. In modern weighing scales, load cells convert the downward force into a proportional electrical signal that can then be converted and displayed to show weight on an indicating element.
When weighing samples in a laboratory, it’s important to remove the added weight of the container, as this can bias the results. This process is called taring, and it can be accomplished by pressing a tare button on the instrument.
Lab Equipment
Measurements of mass are vital for numerous scientific disciplines, including chemistry. The most common method of measuring mass is with a balance, which utilizes an object’s gravitational acceleration to determine its weight. A precise balance is necessary to ensure that all of your measurements are accurate.
For liquid measurements, lab tools like graduated cylinders, pipettes, and burettes provide precise measurements that are vital for accurate chemical reactions. These instruments are engineered with precision in mind, enabling scientists to achieve incredibly small quantities without error.
Other important lab equipment includes a variety of glassware, weighing scales, and heat sources like Bunsen burners and hot plates. Proper storage and organization of all laboratory apparatus helps to ensure that it is safe for use. It also facilitates easy retrieval, reducing time spent searching for the right equipment for an experiment. The right lab apparatus can make or break an experiment, so it is important to carefully assess your needs and select the correct equipment.
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# 1.4: Composition of Functions
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Suppose we wanted to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and the average daily temperature depends on the particular day of the year. Notice how we have just defined two relationships: The temperature depends on the day, and the cost depends on the temperature. Using descriptive variables, we can notate these two functions.
The first function, C(T), gives the cost C of heating a house when the average daily temperature is T degrees Celsius, and the second, T(d), gives the average daily temperature on day d of the year in some city. If we wanted to determine the cost of heating the house on the 5 $${}^{th}$$ day of the year, we could do this by linking our two functions together, an idea called composition of functions. Using the function T(d), we could evaluate T[GrindEQ__5_] to determine the average daily temperature on the 5 $${}^{th}$$ day of the year. We could then use that temperature as the input to the C(T) function to find the cost to heat the house on the 5 $${}^{th}$$ day of the year: C(T(5)).
Definition: Composition of Functions
When the output of one function is used as the input of another, we call the entire operation a composition of functionsComposition of FunctionsFunction:Composition of Functions. We write f(g(x)), and read this as “f of g of x” or “f composed with g at x”.
An alternate notation for composition uses the composition operator: $$\circ$$
$$(f\circ g)(x)$$ is read “f of g of x” or “f composed with g at x”, just like f(g(x)).
Example $$\PageIndex{1}$$
Suppose c(s) gives the number of calories burned doing s sit-ups, and s(t) gives the number of sit-ups a person can do in t minutes. Interpret c(s[GrindEQ__3_]).
When we are asked to interpret, we are being asked to explain the meaning of the expression in words. The inside expression in the composition is s[GrindEQ__3_]. Since the input to the s function is time, the 3 is representing 3 minutes, and s[GrindEQ__3_] is the number of sit-ups that can be done in 3 minutes. Taking this output and using it as the input to the c(s) function will gives us the calories that can be burned by the number of sit-ups that can be done in 3 minutes.
Note that it is not important that the same variable be used for the output of the inside function and the input to the outside function. However, it is essential that the units on the output of the inside function match the units on the input to the outside function, if the units are specified.
Example $$\PageIndex{2}$$
Suppose f(x) gives miles that can be driven in x hours, and g(y) gives the gallons of gas used in driving y miles. Which of these expressions is meaningful: f(g(y)) or g(f(x))?
Solution
The expression g(y) takes miles as the input and outputs a number of gallons. The function f(x) is expecting a number of hours as the input; trying to give it a number of gallons as input does not make sense. Remember the units must match, and number of gallons does not match number of hours, so the expression f(g(y)) is meaningless.
The expression f(x) takes hours as input and outputs a number of miles driven. The function g(y) is expecting a number of miles as the input, so giving the output of the f(x) function (miles driven) as an input value for g(y), where gallons of gas depends on miles driven, does make sense. The expression g(f(x)) makes sense, and will give the number of gallons of gas used, g, driving a certain number of miles, f(x), in x hours.
Exercise $$\PageIndex{1}$$
In a department store you see a sign that says 50% off clearance merchandise, so final cost C depends on the clearance price, p, according to the function C(p). Clearance price, p, depends on the original discount, d, given to the clearance item, p(d). Interpret C(p(d)).
Answer
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Composition of Functions using Tables and GraphsComposition of Functions:Tables and Graphs
When working with functions given as tables and graphs, we can look up values for the functions using a provided table or graph, as discussed in section 1.1. We start evaluation from the provided input, and first evaluate the inside function. We can then use the output of the inside function as the input to the outside function. To remember this, always work from the inside out.
Example $$\PageIndex{1}$$
Using the tables below, evaluate $$f(g(3))$$and $$g(f(4))$$ $63$
Solution
To evaluate $$f(g(3))$$, we start from the inside with the value 3. We then evaluate the inside expression $$g\eqref{GrindEQ__3_}$$using the table that defines the function g: $$g\eqref{GrindEQ__3_}=2$$.
We can then use that result as the input to the f function, so $$g\eqref{GrindEQ__3_}$$ is replaced by the equivalent value 2 and we can evaluate $$f\eqref{GrindEQ__2_}$$. Then using the table that defines the function f, we find that $$f\eqref{GrindEQ__2_}=8$$. $f(g(3))=f(2)=8.$
To evaluate $$g(f(4))$$, we first evaluate the inside expression $$f\eqref{GrindEQ__4_}$$using the first table: $$f\eqref{GrindEQ__4_}=1$$. Then using the table for g we can evaluate: $g(f(4))=g(1)=3.$
Exercise $$\PageIndex{1}$$
2. Using the tables from the example above, evaluate $$f(g(1))$$ and $$g(f(3))$$.
Answer
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Example $$\PageIndex{1}$$
Using the graphs below, evaluate $$f(g(1))$$.
To evaluate $$f(g(1))$$, we again start with the inside evaluation. We evaluate $$g\eqref{GrindEQ__1_}$$ using the graph of the g(x) function, finding the input of 1 on the horizontal axis and finding the output value of the graph at that input. Here, $$g\eqref{GrindEQ__1_}=3$$.
Using this value as the input to the f function, $$f(g(1))=f\eqref{GrindEQ__3_}$$. We can then evaluate this by looking to the graph of the f(x) function, finding the input of 3 on the horizontal axis, and reading the output value of the graph at this input.
$$f\eqref{GrindEQ__3_}=6$$, so $$f(g(1))=6$$.
Exercise $$\PageIndex{1}$$
3. Using the graphs from the previous example, evaluate $$g(f(2))$$.
Answer
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Composition using FormulasComposition of Functions:Formulas
When evaluating a composition of functions where we have either created or been given formulas, the concept of working from the inside out remains the same. First, we evaluate the inside function using the input value provided, then use the resulting output as the input to the outside function.
Example $$\PageIndex{1}$$:
5
Given $$f(t)=t^{2} -t$$ and $$h(x)=3x+2$$, evaluate $$f(h(1))$$.
Since the inside evaluation is $$h\eqref{GrindEQ__1_}$$we start by evaluating the h(x) function at 1: $h(1)=3(1)+2=5$
Then $$f(h(1))=f\eqref{GrindEQ__5_}$$, so we evaluate the f(t) function at an input of 5: $f(h(1))=f(5)=5^{2} -5=20$
Try it Now
4. Using the functions from the example above, evaluate $$h(f(-2))$$.
While we can compose the functions as above for each individual input value, sometimes it would be really helpful to find a single formula which will calculate the result of a composition f(g(x)). To do this, we will extend our idea of function evaluation. Recall that when we evaluate a function like $$f(t)=t^{2} -t$$, we put whatever value is inside the parentheses after the function name into the formula wherever we see the input variable.
Example $$\PageIndex{1}$$:
6
Given $$f(t)=t^{2} -t$$, evaluate $$f\eqref{GrindEQ__3_}$$ and $$f(-2)$$.
$f(3)=3^{2} -3$ $f(-2)=(-2)^{2} -(-2)$
We could simplify the results above if we wanted to $f(3)=3^{2} -3=9-3=6$ $f(-2)=(-2)^{2} -(-2)=4+2=6$
We are not limited, however, to using a numerical value as the input to the function. We can put anything into the function: a value, a different variable, or even an algebraic expression, provided we use the input expression everywhere we see the input variable.
Example $$\PageIndex{1}$$:
7
Using the function from the previous example, evaluate f(a).
This means that the input value for t is some unknown quantity a. As before, we evaluate by replacing the input variable t with the input quantity, in this case a. $f(a)=a^{2} -a$
The same idea can then be applied to expressions more complicated than a single letter.
Example $$\PageIndex{1}$$:
8
Using the same f(t) function from above, evaluate $$f(x+2)$$.
Everywhere in the formula for f where there was a t, we would replace it with the input $$(x+2)$$. Since in the original formula the input t was squared in the first term, the entire input $$x+2$$ needs to be squared when we substitute, so we need to use grouping parentheses. To avoid problems, it is advisable to always use parentheses around inputs.
$f(x+2)=(x+2)^{2} -(x+2)$
We could simplify this expression further to $$f(x+2)=x^{2} +3x+2$$ if we wanted to:
$$f(x+2)=(x+2)(x+2)-(x+2)$$ Use the “FOIL” technique (first, outside, inside, last)
$$f(x+2)=x^{2} +2x+2x+4-(x+2)$$ distribute the negative sign $$f(x+2)=x^{2} +2x+2x+4-x-2$$ combine like terms $f(x+2)=x^{2} +3x+2$
Example $$\PageIndex{9}$$
Using the same function, evaluate $$f(t^{3} )$$.
Solution
Note that in this example, the same variable is used in the input expression and as the input variable of the function. This doesn’t matter – we still replace the original input t in the formula with the new input expression, $$t^{3}$$. $f(t^{3} )=(t^{3} )^{2} -(t^{3} )=t^{6} -t^{3}$
Exercise $$\PageIndex{5}$$
Given $$g(x)=3x-\sqrt{x}$$, evaluate $$g(t-2)$$.
Answer
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This now allows us to find an expression for a composition of functions. If we want to find a formula for f(g(x)), we can start by writing out the formula for g(x). We can then evaluate the function f(x) at that expression, as in the examples above.
Example $$\PageIndex{10}$$
Let $$f(x)=x^{2}$$ and $$g(x)=\frac{1}{x} -2x$$, find f(g(x)) and g(f(x)).
Solution
To find f(g(x)), we start by evaluating the inside, writing out the formula for g(x). $g(x)=\frac{1}{x} -2x$ We then use the expression $$\left(\frac{1}{x} -2x\right)$$ as input for the function f. $f(g(x))=f\left(\frac{1}{x} -2x\right)$
We then evaluate the function f(x) using the formula for g(x) as the input.
Since $$f(x)=x^{2}$$, $$f\left(\frac{1}{x} -2x\right)=\left(\frac{1}{x} -2x\right)^{2}$$
This gives us the formula for the composition: $$f(g(x))=\left(\frac{1}{x} -2x\right)^{2}$$.
Likewise, to find g(f(x)), we evaluate the inside, writing out the formula for f(x) $g(f(x))=g\left(x^{2} \right)$
Now we evaluate the function g(x) using x $${}^{2}$$ as the input. $g(f(x))=\frac{1}{x^{2} } -2x^{2}$
Exercise $$\PageIndex{1}$$
6. Let $$f(x)=x^{3} +3x$$ and $$g(x)=\sqrt{x}$$, find f(g(x)) and g(f(x)).
Answer
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Example $$\PageIndex{11}$$
A city manager determines that the tax revenue, R, in millions of dollars collected on a population of p thousand people is given by the formula $$R(p)=0.03p+\sqrt{p}$$, and that the city’s population, in thousands, is predicted to follow the formula $$p(t)=60+2t+0.3t^{2}$$, where t is measured in years after 2010. Find a formula for the tax revenue as a function of the year.
Solution
Since we want tax revenue as a function of the year, we want year to be our initial input, and revenue to be our final output. To find revenue, we will first have to predict the city population, and then use that result as the input to the tax function. So we need to find R(p(t)). Evaluating this,
$R(p(t))=R\left(60+2t+0.3t^{2} \right)=0.03\left(60+2t+0.3t^{2} \right)+\sqrt{60+2t+0.3t^{2} }$
This composition gives us a single formula which can be used to predict the tax revenue during a given year, without needing to find the intermediary population value.
For example, to predict the tax revenue in 2017, when t = 7 (because t is measured in years after 2010),
$$R(p(7))=0.03\left(60+2(7)+0.3\eqref{GrindEQ__7_}^{2} \right)+\sqrt{60+2\eqref{GrindEQ__7_}+0.3\eqref{GrindEQ__7_}^{2} } \approx 12.079$$million dollars
### Domain of Compositions
When we think about the domain of a composition $$h(x)=f(g(x))$$, we must consider both the domain of the inner function and the domain of the composition itself. While it is tempting to only look at the resulting composite function, if the inner function were undefined at a value of x, the composition would not be possible.
Example $$\PageIndex{12}$$
Let $$f(x)=\frac{1}{x^{2} -1}$$ and $$g(x)=\sqrt{x-2}$$. Find the domain of $$f\left(g(x)\right)$$.
Since we want to avoid the square root of negative numbers, the domain of $$g(x)$$ is the set of values where $$x-2\ge 0$$. The domain is $$x\ge 2$$.
The composition is $$f\left(g(x)\right)=\frac{1}{\left(\sqrt{x-2} \right)^{2} -1} =\frac{1}{(x-2)-1} =\frac{1}{x-3}$$.
The composition is undefined when x = 3, so that value must also be excluded from the domain. Notice that the composition doesn’t involve a square root, but we still have to consider the domain limitation from the inside function.
Combining the two restrictions, the domain is all values of x greater than or equal to 2, except x = 3.
In inequalities, the domain is: $$2\le x<3{\rm \; or\; }x>3$$.
In interval notation, the domain is: $$\left[2,3\right)\cup \left(3,\infty \right)$$.
Exercise $$\PageIndex{1}$$
7. Let $$f(x)=\frac{1}{x-2}$$ and $$g(x)=\frac{1}{x}$$. Find the domain of $$f\left(g(x)\right)$$.
Answer
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### Decomposing Functions
In some cases, it is desirable to decompose a function – to write it as a composition of two simpler functions.
Example $$\PageIndex{13}$$
Write $$f(x)=3+\sqrt{5-x^{2} }$$ as the composition of two functions.
Solution
We are looking for two functions, g and h, so $$f(x)=g(h(x))$$. To do this, we look for a function inside a function in the formula for f(x). As one possibility, we might notice that $$5-x^{2}$$ is the inside of the square root. We could then decompose the function as: $h(x)=5-x^{2}$ $g(x)=3+\sqrt{x}$
We can check our answer by recomposing the functions: $g(h(x))=g\left(5-x^{2} \right)=3+\sqrt{5-x^{2} }$
Note that this is not the only solution to the problem. Another non-trivial decomposition would be $$h(x)=x^{2}$$ and $$g(x)=3+\sqrt{5-x}$$
### Important Topics of this Section
• Definition of Composition of Functions
• Compositions using:
• Words
• Tables
• Graphs
• Equations
• Domain of Compositions
• Decomposition of Functions
Try it Now Answers
1. The final cost, C, depends on the clearance price, p, which is based on the original discount, d. (Or the original discount d, determines the clearance price and the final cost is half of the clearance price.)
2. $$f(g(1))=f\eqref{GrindEQ__3_}=3$$ and $$g(f(3))=g\eqref{GrindEQ__3_}=2$$
3. $$g(f(2))=g\eqref{GrindEQ__5_}=3$$
4. $$h(f(-2))=h\eqref{GrindEQ__6_}=20$$ did you remember to insert your input values using parentheses?
5. $$g(t-2)=3(t-2)-\sqrt{(t-2)}$$
6. $$f(g(x))=f\left(\sqrt{x} \right)=\left(\sqrt{x} \right)^{3} +3\left(\sqrt{x} \right)$$ $$g(f(x))=g\left(x^{3} +3x\right)=\sqrt{\left(x^{3} +3x\right)}$$
1. $$g(x)=\frac{1}{x}$$ is undefined at x = 0.The composition, $$f\left(g(x)\right)=f\left(\frac{1}{x} \right)=\frac{1}{\frac{1}{x} -2} =\frac{1}{\frac{1}{x} -\frac{2x}{x} } =\frac{1}{\frac{1-2x}{x} } =\frac{x}{1-2x}$$ is undefined when $$1-2x=0$$, when $$x=\frac{1}{2}$$. Restricting these two values, the domain is $$\left(-\infty ,0\right)\cup \left(0,\frac{1}{2} \right)\cup \left(\frac{1}{2} ,\infty \right)$$.
## Section 1.4 Exercises
Given each pair of functions, calculate $$f\left(g\left(0\right)\right)$$ and $$g\left(f\left(0\right)\right)$$. $1. f\left(x\right)=4x+8, g\left(x\right)=7-x^{2} 2. f\left(x\right)=5x+7, g\left(x\right)=4-2x^{2}$ $3. f\left(x\right)=\sqrt{x+4} , g\left(x\right)=12-x^{3} 4. f\left(x\right)=\frac{1}{x+2} , g\left(x\right)=4x+3$
$$f(x)$$ $${\it g}({\it x})$$ $$f(x)$$ $${\it g}({\it x})$$Use the table of values to evaluate each expression
1. $$f(g(8))$$
2. $$f\left(g\left(5\right)\right)$$
3. $$g(f(5))$$
4. $$g\left(f\left(3\right)\right)$$
5. $$f(f(4))$$
6. $$f\left(f\left(1\right)\right)$$
7. $$g(g(2))$$
8. $$g\left(g\left(6\right)\right)$$
Use the graphs to evaluate the expressions below.
1. $$f(g(3))$$
2. $$f\left(g\left(1\right)\right)$$
3. $$g(f(1))$$
4. $$g\left(f\left(0\right)\right)$$
5. $$f(f(5))$$
6. $$f\left(f\left(4\right)\right)$$
7. $$g(g(2))$$
8. $$g\left(g\left(0\right)\right)$$
For each pair of functions, find $$f\left(g\left(x\right)\right)$$ and $$g\left(f\left(x\right)\right)$$. Simplify your answers. $21. f\left(x\right)=\frac{1}{x-6} , g\left(x\right)=\frac{7}{x} +6\; 22. f\left(x\right)=\frac{1}{x-4} , g\left(x\right)=\frac{2}{x} +4$ $23. f\left(x\right)=x^{2} +1, g\left(x\right)=\sqrt{x+2} 24. f\left(x\right)=\sqrt{x} +2, g\left(x\right)=x^{2} +3$ $25. f\left(x\right)=\left|x\right|, g\left(x\right)=5x+1 26. f\left(x\right)=\sqrt[{3}]{x} , g\left(x\right)=\frac{x+1}{x^{3} }$
1. If $$f\left(x\right)=\; x^{4} +6$$, $$\; g(x)\; =\; x-6\;$$and $$h(x)\; =\; \sqrt{x}$$, find $$f(g(h(x)))$$
1. If $$f\left(x\right)=\; x^{2} +1$$, $$g\left(x\right)=\frac{1}{x}$$ and $$h\left(x\right)=\; x+3$$ , find $$f(g(h(x)))$$
2. The function $$D(p)$$ gives the number of items that will be demanded when the price is p. The production cost, $$C(x)$$ is the cost of producing x items. To determine the cost of production when the price is \$6, you would do which of the following:
a. Evaluate $$D(C(6))$$ b. Evaluate $$C(D(6))$$
c. Solve $$D(C(x))\; =\; 6$$ d. Solve $$C(D(p))\; =\; 6$$
1. The function $$A(d)$$ gives the pain level on a scale of 0-10 experienced by a patient with d milligrams of a pain reduction drug in their system. The milligrams of drug in the patient’s system after t minutes is modeled by $$m(t)$$. To determine when the patient will be at a pain level of 4, you would need to:
a. Evaluate $$A\left(m\left(4\right)\right)$$ b. Evaluate $$m\left(A\left(4\right)\right)$$
c. Solve $$A\left(m\left(t\right)\right)\; =\; 4$$ d. Solve $$m\left(A\left(d\right)\right)\; =\; 4$$
1. The radius r, in inches, of a spherical balloon is related to the volume, V, by $$r(V)=\sqrt[{3}]{\frac{3V}{4\pi } }$$. Air is pumped into the balloon, so the volume after t seconds is given by $$V\left(t\right)=10+20t$$.
1. Find the composite function $$r\left(V\left(t\right)\right)$$
2. Find the radius after 20 seconds
1. The number of bacteria in a refrigerated food product is given by $$N\left(T\right)=23T^{2} -56T+\; 1\;$$, $$3<T<33$$, where T is the temperature of the food. When the food is removed from the refrigerator, the temperature is given by $$T(t)=5t+1.5$$, where t is the time in hours.
2. Find the composite function $$N\left(T\left(t\right)\right)$$
3. Find the bacteria count after 4 hours
1. Given $$p\left(x\right)=\frac{1}{\sqrt{x} }$$ and $$m\left(x\right)=\; x^{2} -4$$, find the domain of $$m(p(x))\;$$.
34. Given $$p\left(x\right)=\frac{1}{\sqrt{x} }$$ and $$m\left(x\right)=\; 9-x^{2}$$, find the domain of $$m(p(x))\;$$.
35. Given $$f\left(x\right)=\frac{1}{x+3}$$ and $$g\left(x\right)=\frac{2}{x-1}$$, find the domain of $$f\left(g\left(x\right)\right)$$.
36. Given $$f\left(x\right)=\frac{x}{x+1}$$ and $$g\left(x\right)=\frac{4}{x}$$, find the domain of $$f\left(g\left(x\right)\right)$$.
37. Given $$f\left(x\right)=\sqrt{x-2}$$ and $$g\left(x\right)=\frac{2}{x^{2} -3}$$, find the domain of $$g\left(f\left(x\right)\right)$$.
38. Given $$f\left(x\right)=\sqrt{4-x}$$ and $$g\left(x\right)=\frac{1}{x^{2} -2}$$, find the domain of $$g\left(f\left(x\right)\right)$$.
Find functions $$f(x)$$ and $$g(x)$$ so the given function can be expressed as $$h\left(x\right)=f\left(g\left(x\right)\right)$$. $39. h\left(x\right)=\left(x+2\right)^{2} 40. h\left(x\right)=\left(x-5\right)^{3}$ $41. h\left(x\right)=\frac{3}{x-5} 42. h\left(x\right)=\frac{4}{\left(x+2\right)^{2} }$ $43. h\left(x\right)=3+\sqrt{x-2} 44. h\left(x\right)=4+\sqrt[{3}]{x}$ 45. Let $$f(x)$$ be a linear function, with form $$f\left(x\right)=ax+b$$ for constants a and b. [UW]
1. Show that $$f\left(f\left(x\right)\right)$$ is a linear function
2. Find a function $$g(x)$$ such that $$g\left(g\left(x\right)\right)=6x-8$$
46. Let $$f\left(x\right)=\frac{1}{2} x+3$$ [UW]
1. Sketch the graphs of $$f\left(x\right),\; f\left(f\left(x\right)\right),\; f\left(f\left(f\left(x\right)\right)\right)$$ on the interval $$\mathrm{-}$$2 $$\mathrm{\le}$$ x $$\mathrm{\le}$$ 10.
2. Your graphs should all intersect at the point (6, 6). The value x = 6 is called a fixed point of the function f(x)since $$f\eqref{GrindEQ__6_}=6$$; that is, 6 is fixed - it doesn’t move when f is applied to it. Give an explanation for why 6 is a fixed point for any function $$f(f(f(...f(x)...)))$$.
3. Linear functions (with the exception of $$f(x)=x$$) can have at most one fixed point. Quadratic functions can have at most two. Find the fixed points of the function $$g\left(x\right)=x^{2} -2$$.
4. Give a quadratic function whose fixed points are x = $$\mathrm{-}$$2 and x = 3.
47. A car leaves Seattle heading east. The speed of the car in mph after m minutes is given by the function $$C\left(m\right)=\frac{70m^{2} }{10+m^{2} }$$. [UW]
1. Find a function $$m=f(s)$$ that converts seconds s into minutes m. Write out the formula for the new function $$C(f(s))$$; what does this function calculate?
2. Find a function $$m=g(h$$) that converts hours h into minutes m. Write out the formula for the new function $$C(g(h))$$; what does this function calculate?
3. Find a function $$z=v(s)$$ that converts mph s into ft/sec z. Write out the formula for the new function $$v(C(m)$$; what does this function calculate?
$93$
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# complete the following ordered pairs for the equation 2x-5y=17 (1,_) (_,-1) (-4,_)
hala718 | Certified Educator
2x-5y = 17
For the first pair:
(1, _) we are given x-value, then we need to determine y-value:
==> 2x - 5y = 17
==> 2*1 - 5y = 17
==> 2-5y = 17
==> 5y = -15
==> y= -3
==> the pair is (1, -3)
For the second pair: (_,-1)
x-value i unknown and y-value is given -1:
==> 2x - 5y = 17
==> 2x - 5*-1 = 17
==> 2x +5 = 17
==> 2x = 12
==> x= 6
Then the second pair is: (6, -1)
The third pair: (-4, _)
x value is -4, we need y:
2x -5y = 17
2*-4 -5y = 17
-8 - 5y = 17
-5y = 25
==> y= -5
Then the third pair is (-4, -5)
neela | Student
2x-5y = 17. To complete the ordered pairs which satisfy the equation: (1, ...), (..., -1) , (-4 , ...)
We substitut the one coordinate inthe respective variable of the equation and solve for the other variable.
(1 , ... ) : We substitute x = 1 , in 2x-5y=17 and find the y vaue:
2*1-5y = 17. Or -5y =17-2 = 15. Or y = 15/-5 = -3
(... , -1): We substitute y = -1 in 2x-5y = 17 and solve for x: (2x-5(-1) = -19. So 2x +5 = 17. Or 2x = 17-5 = 12. Or x = 12/2 = 6.
((-4, ...): To find the missing y coordinate we put x = -4 in 2x-5y = 17 and solve for x. 2(-4)-5y = 17. Or -8-5y = 17. Or -5y = 17+8=25. Or y = 25/-5 = -5.
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Maximum number of envelopes that can be put inside other bigger envelopes
Given N number of envelopes, as {W, H} pair, where W as the width and H as the height. One envelope can fit into another if and only if both the width and height of one envelope is greater than the width and height of the other envelope. Find the maximum number of envelopes that can be put inside another envelope and so on. Rotation of envelope is not allowed.
Examples:
Input: envelope[] = {{4, 3}, {5, 3}, {5, 6}, {1, 2}}
Output: 3
Explanation:
The maximum number of envelopes that can be put into another envelope
is 3.
({1, 2}, {4, 3}, {5, 6})
Input: envelope[] = {{3, 6}, {5, 4}, {4, 8}, {6, 9}, {10, 7}, {12, 12}}
Output: 4
Explanation:
The maximum number of envelopes that can be put into another envelope is 4.
({3, 6}, {4, 8}, {6, 9}, {12, 12})
Naive Approach: This problem is similar to the Longest Increasing Subsequence problem of Dynamic Programming. The idea is to sort the envelopes in non-decreasing order and for each envelope check the number of envelopes that can be put inside that envelope. Follow the steps below to solve the problem:
• Sort the array in the non-decreasing order of width and height.
• Initialize a dp[] array, where dp[i] stores the number of envelopes that can be put inside with envelope[i] as the largest envelope.
• For each envelope[i], loop through the envelopes smaller than itself and check if the width and the height of the smaller envelope is strictly less than that of envelope[i]. If it is less, than the smaller envelope can be put inside envelope[i].
• The maximum of the dp[] array gives the maximum number of envelopes that can be put inside one another.
Below is the implementation of the above approach:
C++
// C++ program for the above approach#include using namespace std; // Function that returns the maximum// number of envelopes that can be// inserted into another envelopesint maxEnvelopes(vector > envelopes){ // Number of envelopes int N = envelopes.size(); if (N == 0) return N; // Sort the envelopes in // non-decreasing order sort(envelopes.begin(), envelopes.end()); // Initialize dp[] array int dp[N]; // To store the result int max_envelope = 1; dp[0] = 1; // Loop through the array for (int i = 1; i < N; ++i) { dp[i] = 1; // Find envelopes count for // each envelope for (int j = 0; j < i; ++j) { if (envelopes[i][0] > envelopes[j][0] && envelopes[i][1] > envelopes[j][1] && dp[i] < dp[j] + 1) dp[i] = dp[j] + 1; } // Store maximum envelopes count max_envelope = max(max_envelope, dp[i]); } // Return the result return max_envelope;} // Driver Codeint main(){ // Given the envelopes vector > envelopes = { { 4, 3 }, { 5, 3 }, { 5, 6 }, { 1, 2 } }; // Function Call cout << maxEnvelopes(envelopes); return 0;}
Java
// Java program for the above approachimport java.util.*;import java.lang.*; class GFG{ // Function that returns the maximum// number of envelopes that can be// inserted into another envelopesstatic int maxEnvelopes(int[][] envelopes){ // Number of envelopes int N = envelopes.length; if (N == 0) return N; // Sort the envelopes in // non-decreasing order Arrays.sort(envelopes, (a, b) -> (a[0] != b[0]) ? a[0] - b[0] : a[1] - b[1]); // Initialize dp[] array int[] dp = new int[N]; // To store the result int max_envelope = 1; dp[0] = 1; // Loop through the array for(int i = 1; i < N; ++i) { dp[i] = 1; // Find envelopes count for // each envelope for(int j = 0; j < i; ++j) { if (envelopes[i][0] > envelopes[j][0] && envelopes[i][1] > envelopes[j][1] && dp[i] < dp[j] + 1) dp[i] = dp[j] + 1; } // Store maximum envelopes count max_envelope = Math.max(max_envelope, dp[i]); } // Return the result return max_envelope;} // Driver Codepublic static void main (String[] args){ // Given the envelopes int[][] envelopes = { { 4, 3 }, { 5, 3 }, { 5, 6 }, { 1, 2 } }; // Function call System.out.println(maxEnvelopes(envelopes));}} // This code is contributed by offbeat
Python3
# Python3 program for the above approach # Function that returns the maximum# number of envelopes that can be# inserted into another envelopesdef maxEnvelopes(envelopes): # Number of envelopes N = len(envelopes) if (N == 0): return N # Sort the envelopes in # non-decreasing order envelopes = sorted(envelopes) # Initialize dp[] array dp = [0] * N # To store the result max_envelope = 1 dp[0] = 1 # Loop through the array for i in range(1, N): dp[i] = 1 # Find envelopes count for # each envelope for j in range(i): if (envelopes[i][0] > envelopes[j][0] and envelopes[i][1] > envelopes[j][1] and dp[i] < dp[j] + 1): dp[i] = dp[j] + 1 # Store maximum envelopes count max_envelope = max(max_envelope, dp[i]) # Return the result return max_envelope # Driver Codeif __name__ == '__main__': # Given the envelopes envelopes = [ [ 4, 3 ], [ 5, 3 ], [ 5, 6 ], [ 1, 2 ] ] # Function Call print(maxEnvelopes(envelopes)) # This code is contributed by Mohit Kumar
C#
// C# program to implement above approachusing System;using System.Collections;using System.Collections.Generic; class GFG{ // Function that returns the maximum // number of envelopes that can be // inserted into another envelopes static int maxEnvelopes(int[][] envelopes) { // Number of envelopes int N = envelopes.Length; if (N == 0){ return N; } // Sort the envelopes in // non-decreasing order Array.Sort(envelopes, new comp()); // Initialize dp[] array int[] dp = new int[N]; // To store the result int max_envelope = 1; dp[0] = 1; // Loop through the array for(int i = 1 ; i < N ; ++i) { dp[i] = 1; // Find envelopes count for // each envelope for(int j = 0 ; j < i ; ++j) { if (envelopes[i][0] > envelopes[j][0] && envelopes[i][1] > envelopes[j][1] && dp[i] < dp[j] + 1){ dp[i] = dp[j] + 1; } } // Store maximum envelopes count max_envelope = Math.Max(max_envelope, dp[i]); } // Return the result return max_envelope; } // Driver code public static void Main(string[] args){ // Given the envelopes int[][] envelopes = new int[][]{ new int[]{ 4, 3 }, new int[]{ 5, 3 }, new int[]{ 5, 6 }, new int[]{ 1, 2 } }; // Function call Console.WriteLine(maxEnvelopes(envelopes)); }} class comp : IComparer{ public int Compare(int[] a, int[] b) { if(a[0] != b[0]) return a[0] - b[0]; return a[1] - b[1]; }} // This code is contributed by entertain2022.
Javascript
Output:
3
Time Complexity: O(N2)
Auxiliary Space: O(N)
Efficient Approach:To optimize the naive approach the idea is to use the concept of Binary Search and Longest Increasing Subsequence. Sorting the envelopes in the increasing order of width and the decreasing order of height if width is same, reduces the problem to finding the longest increasing sequence of height of the envelope. This approach works as width is already sorted in increasing order and only maximum increasing sequence of height is sufficient to find the maximum number of envelopes. The efficient way to find the Longest Increasing Sequence in N×log(N) approach is discussed in this article.
Below is the implementation of the above approach:
C++
// C++ program for the above approach#include using namespace std; // Function that returns the maximum// number of envelopes that can be// inserted into another envelopesint maxEnvelopes(vector >& envelopes){ // Number of envelopes int N = envelopes.size(); if (N == 0) return N; // Sort the envelopes in increasing // order of width and decreasing order // of height is width is same sort(envelopes.begin(), envelopes.end(), [](vector& a, vector& b) { return a[0] < b[0] or (a[0] == b[0] and a[1] > b[1]); }); // To store the longest increasing // sequence of height vector dp; // Finding LIS of the heights // of the envelopes for (int i = 0; i < N; ++i) { auto iter = lower_bound(dp.begin(), dp.end(), envelopes[i][1]); if (iter == dp.end()) dp.push_back(envelopes[i][1]); else if (envelopes[i][1] < *iter) *iter = envelopes[i][1]; } // Return the result return dp.size();} // Driver Codeint main(){ // Given the envelopes vector > envelopes = { { 4, 3 }, { 5, 3 }, { 5, 6 }, { 1, 2 } }; // Function Call cout << maxEnvelopes(envelopes); return 0;}
Java
// Java program for the above approachimport java.io.*;import java.util.*;import java.util.Arrays;import java.util.Collections; class GFG{ // Function that returns the maximum // number of envelopes that can be // inserted into another envelopes static int maxEnvelopes(int[][] envelopes) { // Number of envelopes int N = envelopes.length; if (N == 0) return N; // Sort the envelopes in increasing // order of width and decreasing order // of height is width is same Arrays.sort(envelopes,new Comparator() { @Override public int compare(int[] a, int[] b) { return a[0] == b[0] ? b[1] - a[1] : a[0] - b[0];; } }); // To store the longest increasing // sequence of height ArrayList dp = new ArrayList(); // Finding LIS of the heights // of the envelopes for (int i = 0; i < N; ++i) { int iter = Collections.binarySearch(dp, envelopes[i][1]); if (iter < 0) iter=Math.abs(iter)-1; if(iter == dp.size()) dp.add(envelopes[i][1]); else if (envelopes[i][1] < dp.get(iter)) dp.set(iter,envelopes[i][1]); } // Return the result return dp.size(); } // Driver Code public static void main (String[] args) { // Given the envelopes int[][] envelopes = { { 4, 3 }, { 5, 3 }, { 5, 6 }, { 1, 2 } }; // Function Call System.out.println(maxEnvelopes(envelopes)); }} // This code is contributed by Aman Kumar
Python3
# Python program for the above approachfrom bisect import bisect_left as lower_bound # Function that returns the maximum# number of envelopes that can be# inserted into another envelopesdef maxEnvelopes(envelopes): # Number of envelopes N = len(envelopes) if(N == 0): return N # Sort the envelopes in increasing # order of width and decreasing order # of height is width is same envelopes.sort() # To store the longest increasing # sequence of height dp = [] # Finding LIS of the heights # of the envelopes for i in range(N): iter = lower_bound(dp,envelopes[i][1]) if(iter == len(dp)): dp.append(envelopes[i][1]) elif(envelopes[i][1] < dp[iter]): dp[iter] = envelopes[i][1] # Return the result return len(dp) # Driver Code # Given the envelopesenvelopes = [[4, 3], [5, 3], [5, 6], [1, 2]] # Function Callprint(maxEnvelopes(envelopes)) # This code is contributed by Pushpesh Raj
C#
using System;using System.Linq;using System.Collections.Generic; class GFG{ // Function that returns the maximum // number of envelopes that can be // inserted into another envelopes static int maxEnvelopes(int[][] envelopes) { // Number of envelopes int N = envelopes.Length; if (N == 0) return N; // Sort the envelopes in increasing // order of width and decreasing order // of height is width is same Array.Sort(envelopes, (a, b) = > a[0] - b[0]); // To store the longest increasing // sequence of height List dp = new List(); // Finding LIS of the heights // of the envelopes for (int i = 0; i < N; ++i) { int iter = dp.BinarySearch(envelopes[i][1]); if (iter < 0) iter = ~iter; if (iter == dp.Count) dp.Add(envelopes[i][1]); else if (envelopes[i][1] < dp[iter]) dp[iter] = envelopes[i][1]; } // Return the result return dp.Count; } // Driver Code static void Main(string[] args) { // Given the envelopes int[][] envelopes = new int[4][]; envelopes[0] = new int[] { 4, 3 }; envelopes[1] = new int[] { 5, 3 }; envelopes[2] = new int[] { 5, 6 }; envelopes[3] = new int[] { 1, 2 }; // Function Call Console.WriteLine(maxEnvelopes(envelopes)); }} // This code is contributed by lokeshpotta20.
Javascript
// JavaScript program for the above approach // Function that returns the maximum// number of envelopes that can be// inserted into another envelopesfunction maxEnvelopes(envelopes) { // Number of envelopes let N = envelopes.length; if (N === 0) return N; // Sort the envelopes in increasing // order of width and decreasing order // of height is width is same envelopes.sort((a, b) => { if (a[0] === b[0]) { return b[1] - a[1]; } else { return a[0] - b[0]; } }); // To store the longest increasing // sequence of height let dp = []; // Finding LIS of the heights // of the envelopes for (let i = 0; i < N; i++) { let iter = dp.findIndex(x => x >= envelopes[i][1]); if (iter === -1) { dp.push(envelopes[i][1]); } else if (envelopes[i][1] < dp[iter]) { dp[iter] = envelopes[i][1]; } } // Return the result return dp.length;} // Driver Codelet envelopes = [[4, 3], [5, 3], [5, 6], [1, 2]]; // Function Callconsole.log(maxEnvelopes(envelopes));// this contributed by devendra
Output:
3
Time Complexity: O(N*log(N))
Auxiliary Space: O(N)
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PDA
View Full Version : Another course 2 question
bg23516
04-21-2002, 04:15 PM
Company A invests \$10,000 in machinery that is expected to yield net cash flows of \$7000 in each of the next 2 years. Company A intends to record depreciation costs of \$5000 in each of the next 2 years.
Find the Economic Rate of Return and Book Rate of Return on Investment if cost of capital is 10%.
---
I understand how to find the the economic rate of return....
Economic Income = CF1 + PV1 - PV0 = 1215
EROR = EI/PV0 = 10%
My problem is with finding book ROI...
BROI = Book Income / BV0 where:
Book Income = CF1 + BV1 - BV0
According to the solutions, BV0 = 10,000 (2*5000) and BV1 = 5000. Why is book value equal to the depreciation, and why is it just added, despite the different years?
Gandalf
04-21-2002, 04:49 PM
Why am I trying to discuss course 2 having none of the study material?
That said, I suspect BV0 = 10,000 because that's what you paid for it. It's somewhat of a coincidence that it equals 2 * 5,000, but not completely coincidence. It is true that BV0 = sum of depreciation in all future periods + any residual value after the depreciation, and that in many situations such as this one residual value (at least residual book value) is 0.
Normally BV1 does not equal depreciation. BV1 = BV0 - depreciation during 1. Here, BV1 = 10,000 - depreciation just happens to equal depreciation.
Those are my guesses, anyway.
ASA_Woman
04-21-2002, 06:39 PM
Gandalf is correct. BV0 is the initial value of the machinery, ie 10,000. BV1 is the value at the end of 1 year, ie after one year's worth of depreciation. So BV1 = BV0 - 5,000 = 10,000 - 5,000 = 5,000.
The question states that the depreciation is 5,000 each year. It just works out this time that BV0 is 2 times the depreciation. That will not always be the case.
GuyInWestGrove
04-12-2004, 03:35 PM
Sorry to resurrect this thread two years later, but ..
This appears to be problem 47 from the sample exam (Adobe PDF at http://casact.org/admissions/studytools/exam2/sampleExam2.pdf
The original post seems to indicate that the economic income calculation in the solution was in error. The solution shows CF = 7, PV at beginning of year = 10, PV at EOF = 7/1.1 = 6.36. Economic income = 7 - (10-6.36) = 3.36. Return = 3.36 / 10.
Calculating the PV's from the discounted future cash flows as bg23516 has jives with what Braeley and Meyers seem to do.
A couple of questions --
1. Is the solution in the PDF correct?
2. How would you know to look at the return in only the first year, and not over say both years, or even just the second year?
3. What is the origin of the sample exam? (i.e. Did these questions appear on real exams in the past? They seem harder than the other published exams.)
Svak
04-13-2004, 12:58 AM
B&M are referring to year when calculating Rate of Return either Economic Rate of Return or Book Rate of Return.
If we assume question is on return during the "first year", then,
Economic Rate of Return
Change in PV (from year 0 to year 1) /PV at year 0
[CF<sub>1</sub>+ PV<sub>1</sub>- PV<sub>0</sub>]÷PV<sub>0</sub>
[7000+6363.63-12147.70]÷12147.70
=10%
Solution is different from this that it is using 10000 as PV<sub>0</sub>
It makes sense to take 10000 in the denominator, since this is what we are investing. But is it the Present Value at time zero?
Book Rate of Return
Book Rate Return, I think is straight forward. There is no confusion.
[CF<sub>1</sub>+BV<sub>1</sub>-BV<sub>0</sub>]÷BV<sub>0</sub>
[7000+5000-10000]÷10000
Investing 10000 and receiving 2000 net income (7000 cash flow and losing 5000 as depreciation).
Gandalf
04-13-2004, 08:47 AM
B&M is referring to year when calculating Rate of Return either Economic Rate of Return or Book Rate of Return.
If we assume question is on return during the "first year", then,
Economic Rate of Return
Change in PV (from year 0 to year 1) /PV at year 0
[CF<sub>1</sub>+ PV<sub>1</sub>- PV<sub>0</sub>]÷PV<sub>0</sub>
[7000+6363.63-12147.70]÷12147.70
=10%
Solution is different from this that it is using 10000 as PV<sub>0</sub>
It makes sense to take 10000 in the denominator, since this is what we are investing. But is it the Present Value at time zero?
You really need to see if the text specifies what to do. One thing that seems clear to me is that you must use the same value for PV0 in both numerator and denominator.
Which to use depends on what is meant by "first year" (and the problem itself didn't even specify "first year" :swear: )
If the first year is from the time you had 10,000 and ends 1 year later, then PV0 should be 10,000. You started with 10,000, and added economic value during the period by making the investment.
If the first year is one year starting immediately after the moment the investment was made, then the economic gain at the moment of investment is before the first year, and PV0 should be 12,148.
Becoming Actuary
04-14-2004, 10:20 AM
In book rate of return -
The initial value BV0 is always equal to initial investment (10,000). and book value at time 1, BV1 is BV0-depr (which in this case is 5000. sometimes they say its a st. line depr. for lets just say 5 years then you calculate the depr = initial invest./#of yrs.=10,000/5 = 2000).
san
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## 5+5= 9+3= 20+15= 25+38=
Question
5+5=
9+3=
20+15=
25+38=
in progress 0
22 hours 2021-07-21T20:43:58+00:00 2 Answers 0 views 0
1. 5+5=10
9+3=12
20+15=35
25+38=63
2. I think it’s
=10
=12
=35
=63
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Test 3 Section 3 #20
To get this one, first draw triangle ABC with the information given: angle B is a right angle,
BC = 16, and AC = 20.
Because you know your Pythagorean triples, you know that this is a big cousin of the 3-4-5 triangle—it’s a 12-16-20!
Now, let me point out something that you may already have realized: if triangle DEF is similar to triangle ABC, that means it has the same angles! Since the sine of an angle is always the same ratio regardless of the lengths of the sides in any particular triangle, we don’t need to draw DEF or calculate its side lengths to know what the sine of angle F is! The question tells us that angle F corresponds to angle C, so sin F = sin C. All we need to do is calculate the sine of C. So, SOH-CAH-TOA that bad boy.
Note that you can grid the fraction, 3/5, or the decimal equivalent .6 and be correct.
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# Physics Formulae/Conservation and Continuity Equations
Lead Article: Tables of Physics Formulae
To summarize essentials of physics, this section enumerates the classical conservation laws and continuity equations. All the following conservation laws carry through to modern physics, such as Quantum Mechanics, Relativity, Particle Physics and Quantum Relativity, though modifications to conserved quantities may be necessary. Particle physics introduces new conservation laws, many in a different way using quantum numbers.
For any isolated system (i.e. independent of external agents/influences) the following laws apply to the whole system. Constituents of the system possessing these quantities may experience changes, but the total amount of the quantity due to all constituents is constant.
Two equivalent ways of applying these in problems is by considering the quantities before and after an event, or considereing any two points in space and time, and equating the initial state of the system to the final, since the quantity is conserved.
Corresponding to conserved quantities are currents, current densities, or other time derivatives. These quantites must be conserved also since the amount of a conserved quantity associated with a system is invariant in space and time.
## Classical Conservation
Conserved Quantity Constancy Equation System Equation/s Time Derivatives
Mass ${\displaystyle \Delta m=0\,\!}$ ${\displaystyle M_{\mathrm {system} }=\sum _{i=1}^{N_{1}}m_{i}=\sum _{j=1}^{N_{2}}m_{j}\,\!}$ Mass current conservation
${\displaystyle \sum _{i=1}^{N_{1}}\left(I_{\mathrm {m} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(I_{\mathrm {m} }\right)_{j}=0\,\!}$
${\displaystyle \sum _{i=1}^{N_{1}}\left(\mathbf {j} _{\mathrm {m} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(\mathbf {j} _{\mathrm {m} }\right)_{j}=\mathbf {0} \,\!}$
Linear Momentum ${\displaystyle \Delta \mathbf {p} =\mathbf {0} \,\!}$ ${\displaystyle \sum _{i=1}^{N_{1}}\mathbf {p} _{i}=\sum _{i=1}^{N_{2}}\mathbf {p} _{j}\,\!}$
which can be written in equivalent ways, most useful forms are:
${\displaystyle \sum _{i=1}^{N_{1}}m_{i}\mathbf {v} _{i}=\sum _{j=1}^{N_{2}}m_{j}\mathbf {v} _{j}}$
Momentum current conservation
${\displaystyle \sum _{i=1}^{N_{1}}\left(I_{\mathrm {p} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(I_{\mathrm {p} }\right)_{j}=0\,\!}$
Momentum current density conservation
${\displaystyle \sum _{i=1}^{N_{1}}\left(\mathbf {j} _{\mathrm {p} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(\mathbf {j} _{\mathrm {p} }\right)_{j}=\mathbf {0} \,\!}$
Total Angular Momentum ${\displaystyle \Delta \mathbf {L} _{\mathrm {total} }=\mathbf {0} \,\!}$ ${\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}\mathbf {L} _{i}=\sum _{j=1}^{N_{2}}\mathbf {L} _{j}\,\!}$
which can be written in equivalent ways, most useful forms are:
${\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}\left(\mathbf {I} _{\mathrm {ab} }{\boldsymbol {\omega }}_{\mathrm {b} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(\mathbf {I} _{\mathrm {ab} }{\boldsymbol {\omega }}_{\mathrm {b} }\right)_{j}\,\!}$
${\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}\mathbf {r} _{i}\times \mathbf {p} _{i}=\sum _{j=1}^{N_{2}}\mathbf {r} _{j}\times \mathbf {p} _{j}\,\!}$
${\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)=\sum _{j=1}^{N_{2}}m_{j}\left(\mathbf {r} _{j}\times \mathbf {v} _{j}\right)\,\!}$
No analogue
Spin Angular Momentum ${\displaystyle \Delta \mathbf {L} _{\mathrm {spin} }=\mathbf {0} \,\!}$ Same as above
Orbital Angular Momentum ${\displaystyle \Delta \mathbf {L} _{\mathrm {orbital} }=\mathbf {0} \,\!}$ Same as above
Energy ${\displaystyle \Delta E=0\,\!}$ ${\displaystyle E_{\mathrm {system} }=\sum _{i}T_{i}+\sum _{j}V_{j}\,\!}$
or simply
${\displaystyle E=T+V\,\!}$
${\displaystyle E_{\mathrm {system} }=\sum _{i=1}^{N_{1}}\left(T_{i}+V_{i}\right)=\sum _{j=1}^{N_{2}}\left(T_{j}+V_{j}\right)\,\!}$
Power conservation
${\displaystyle \sum _{i}P_{i}+\sum _{j}P_{j}=0\,\!}$
Intensity conservation
${\displaystyle \sum _{i}I_{i}+\sum _{j}I_{j}=0\,\!}$
Charge ${\displaystyle \Delta q=0\,\!}$ ${\displaystyle Q_{\mathrm {system} }=\sum _{i=1}^{N_{1}}q_{i}=\sum _{j=1}^{N_{2}}q_{j}\,\!}$ Electric current conservation
${\displaystyle \sum _{i=1}^{N_{1}}I_{i}=\sum _{j=1}^{N_{2}}I_{j}=0\,\!}$
Electric current density conservation
${\displaystyle \sum _{i=1}^{N_{1}}\mathbf {J} _{i}=\sum _{j=1}^{N_{2}}\mathbf {J} _{j}=\mathbf {0} \,\!}$
## Classical Continuity Equations
Continuity equations describe transport of conserved quantities though a local region of space. Note that these equations are not fundamental simply because of conservation; they can be derived.
Continuity Description Nomenclature General Equation Simple Case
Hydrodynamics, Fluid Flow ${\displaystyle j_{\mathrm {m} }\,\!}$ = Mass current current at the cross-section
${\displaystyle \rho \,\!}$ = Volume mass density
${\displaystyle \mathbf {u} \,\!}$ = velocity field of fluid
${\displaystyle \mathbf {A} \,\!}$ = cross-section
${\displaystyle \nabla \cdot (\rho \mathbf {u} )+{\partial \rho \over \partial t}=0\,\!}$ ${\displaystyle j_{\mathrm {m} }=\rho _{1}\mathbf {A} _{1}\cdot \mathbf {u} _{1}=\rho _{2}\mathbf {A} _{2}\cdot \mathbf {u} _{2}\,\!}$
Electromagnetism, Charge ${\displaystyle I\,\!}$ = Electric current at the cross-section
${\displaystyle \mathbf {J} \,\!}$ = Electric current density
${\displaystyle \rho \,\!}$ = Volume electric charge density
${\displaystyle \mathbf {u} \,\!}$ = velocity of charge carriers
${\displaystyle \mathbf {A} \,\!}$ = cross-section
${\displaystyle \nabla \cdot \mathbf {J} +{\partial \rho \over \partial t}=0\,\!}$ ${\displaystyle I=\rho _{1}\mathbf {A} _{1}\cdot \mathbf {u} _{1}=\rho _{2}\mathbf {A} _{2}\cdot \mathbf {u} _{2}\,\!}$
Quantum Mechnics, Probability ${\displaystyle \mathbf {j} \,\!}$ = probability current/flux
${\displaystyle P=P(x,t)\,\!}$ = probability density function
${\displaystyle \nabla \cdot \mathbf {j} +{\frac {\partial P}{\partial t}}=0\,\!}$
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In the Below Fig, Aoc is a Line, Find X. - Mathematics
In the below fig, AOC is a line, find x.
Solution
Since ∠ AOB and ∠BOC are linear pairs
∠AOB + ∠BOC = 180°
⇒70° + 2x° = 180°
⇒ 2x° = 180° - 70°
⇒ 2x = 110°
⇒ x 110/2
⇒ x = 55°
Concept: Pairs of Angles
Is there an error in this question or solution?
Chapter 10: Lines and Angles - Exercise 10.2 [Page 15]
APPEARS IN
RD Sharma Mathematics for Class 9
Chapter 10 Lines and Angles
Exercise 10.2 | Q 9 | Page 15
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# If each member of a 10 person class earned an integer score between 1
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If each member of a 10 person class earned an integer score between 1 and 10, inclusive, on a recent quiz and the range between the highest and lowest scores was 8, what is the lowest average score possible for the class?
A. 1.2
B. 1.8
C. 2
D. 2.6
E. 4.5
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Re: If each member of a 10 person class earned an integer score between 1 [#permalink]
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20 Jun 2018, 00:01
Bunuel wrote:
If each member of a 10 person class earned an integer score between 1 and 10, inclusive, on a recent quiz and the range between the highest and lowest scores was 8, what is the lowest average score possible for the class?
A. 1.2
B. 1.8
C. 2
D. 2.6
E. 4.5
As we're asked about 'the lowest possible value' we'll look at the extremes.
We're looking for the lowest possible average so we want as many students as possible to have the minimum value - 1.
Since 1 studnet must have a score of 1 + 8 = 9, this means that we can give 9 students a score of 1 and 1 a score of 9.
This sums to 1*9 +9*1 = 18.
Dividing by the number of students gives the average: 18/10 = 1.8
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20 Jun 2018, 02:59
Solution
Given:
$$• There are 10 persons in a class. • The score of each person of the class is between 1 and 10, inclusive. • The range of the score earned by the all the 10 persons is 8. To find: • We need to find the the lowest averagSe score possible for the class.$$
Approach and Working:
We will get the lowest average of the class if all the persons score the lowest possible score.
Between 1 to 10, the lowest score is 1."]"
Hence, nine students of the class scores 1 and one student scores 9 as range must be 9.
• Thus, average= 9*1+1*9/10= 18/10= 1.8
Hence, option B is the correct answer."
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If each member of a 10 person class earned an integer score between 1 [#permalink]
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21 Jul 2018, 09:30
EgmatQuantExpert wrote:
Solution
Given:
$$• There are 10 persons in a class. • The score of each person of the class is between 1 and 10, inclusive. • The range of the score earned by the all the 10 persons is 8. To find: • We need to find the the lowest averagSe score possible for the class.$$
Approach and Working:
We will get the lowest average of the class if all the persons score the lowest possible score.
Between 1 to 10, the lowest score is 1."]"
Hence, nine students of the class scores 1 and one student scores 9 as range must be 9.
• Thus, average= 9*1+1*9/10= 18/10= 1.8
Hence, option B is the correct answer."
Thanks for the explanation but the question says that each student in the class earned an INTEGER score... From this line I expected that even the extremes of the scores should be integers. Or am I wrong?
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21 Jul 2018, 10:13
1
Kem12 wrote:
Thanks for the explanation but the question says that each student in the class earned an INTEGER score... From this line I expected that even the extremes of the scores should be integers. Or am I wrong?
Posted from my mobile device
Hi Kem12,
I think that you are confused by this: average= 9*1+1*9/10= 18/10= 1.8. Indeed it should be average= (9*1+1*9)/10= 18/10= 1.8. The second extreme is not a fraction.
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Joined: 18 Apr 2018
Posts: 93
Re: If each member of a 10 person class earned an integer score between 1 [#permalink]
### Show Tags
22 Jul 2018, 00:56
Hero8888 wrote:
Kem12 wrote:
Thanks for the explanation but the question says that each student in the class earned an INTEGER score... From this line I expected that even the extremes of the scores should be integers. Or am I wrong?
Posted from my mobile device
Hi Kem12,
I think that you are confused by this: average= 9*1+1*9/10= 18/10= 1.8. Indeed it should be average= (9*1+1*9)/10= 18/10= 1.8. The second extreme is not a fraction.
Thanks. I now understand.
Re: If each member of a 10 person class earned an integer score between 1 [#permalink] 22 Jul 2018, 00:56
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# Conductor in Capacitor
1. Sep 17, 2008
### voelkner
1. The problem statement, all variables and given/known data
An isolated capacitor with capacitance C = 1 µF has a charge Q = 29 µC on its plates
A conductor is inserted into the capacitor with thickness of the conductor is 1/3 the thickness of the capacitor and is centered in between the plates of the capacitor.
What is the capacitance of the capacitor with the conductor in place???
2. Relevant equations
C = Q/V
V = E*d
3. The attempt at a solution
I've been trying this problem for hours. I know that since the capacitor is a conductor it makes the distance between the two plates smaller which means that the capacitance should therefore increase. I thought that since we now had two distances that were each 1/3 the original distance the capacitance would increase by a factor of 6 however this answer does not work. Can anyone help me?!?
2. Sep 17, 2008
### tiny-tim
Hi voelkner!
6 = 3 + 3 … isn't that for capacitors in parallel?
these capacitors are in series.
3. Sep 17, 2008
### voelkner
Would it then be 1/C?
4. Sep 17, 2008
### tiny-tim
1/C = 1/C1 + 1/C2
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# 2.8: Number Properties
Difficulty Level: At Grade Created by: CK-12
## Introduction
The Week of Sales
Once the school store had been up and running for a few weeks, the students saw what a huge success it was. Students began to rely on getting their school supplies and school and the student council was overjoyed at the amount of money that they would have for events.
“We are doing terrific!” Kelly said at the meeting one week.
“Yes. Here are the total sales that we have done for the month.” Trevor said taking out his paper. “We made 130.25, 75.18, and 85.00 for a grand total of 380 dollars and 43 cents.”
“Wait a minute, there are four weeks in the month. You only said three amounts,” Mallory pointed out.
Trevor looked back down at his paper.
“You’re right. I missed week 3. But I know the total is correct.”
This is where you come in. Based on the information that Trevor gave the group, you should be able to figure out how much money was made in the missing week. This lesson will show you how to use properties to work with variable and numerical expressions. At the end of this lesson, you will write an equation and use properties to figure out the sales for the third week.
What You Will Learn
In this lesson, you will learn to use the following skills.
• Identify and apply the associative property of addition and multiplication in rational number operations, using numerical and variable expressions.
• Identify and apply the commutative property of addition and multiplication in rational number operations, using numerical and variable expressions.
• Identify and apply the distributive property in rational number operations, using numerical and variable expressions.
• Model and solve real-world problems using equations involving rational numbers.
Teaching Time
In this chapter, you have learned about all different kinds of rational numbers and the different number properties. It’s important to know how and when to use these properties when given a real-world situation. Let’s take a look at how properties can help you solve problems.
First, let’s review all of the properties you have learned.
The grouping of addends does not affect the sum: \begin{align*}4.5+(2.1+9.6)=(4.5+2.1)+9.6\end{align*}
Associative Property of Multiplication
The grouping of numbers does not affect the product: \begin{align*}4.5 \times (2.1 \times 9.6)=(4.5 \times 2.1) \times 9.6\end{align*}
The order of addends does not change the sum: \begin{align*}6.3+8.7=8.7+6.3\end{align*}
Commutative Property of Multiplication
The order of numbers does not change the product: \begin{align*}6.3 \times 8.7 = 8.7 \times 6.3\end{align*}
Distributive Property
The product of a number and a sum is equal to the sum of the individual products of addends and the number: \begin{align*}3.2 (1.5 + 8.9)=(3.2 \times 1.5)+(3.2 \times 8.9)\end{align*}
The sum of any number and zero is that number: \begin{align*}\frac{3}{11}+0=\frac{3}{11}\end{align*}
The sum of any number and its inverse is zero: \begin{align*}\frac{3}{4}+ \left(-\frac{3}{4}\right)=0\end{align*}
Multiplicative Identity
The product of any number and one is that number: \begin{align*}\frac{3}{11} \times 1=\frac{3}{11}\end{align*}
Multiplicative Inverse
The product of any number and its reciprocal is one: \begin{align*}\frac{3}{4} \times\frac{4}{3}=1\end{align*}
Zero Property
The product of any number and zero is zero: \begin{align*}\frac{4}{7} \times 0=0\end{align*}
The Order of Operations
• First evaluate expressions in parentheses.
• Then evaluate exponents.
• Then multiply and divide in order from left to right.
• Finally, add and subtract in order from left to right.
I. Identify and Apply the Associative Property of Addition and Multiplication in Rational Number Operations, using Numerical and Variable Expressions
We can start by remembering that when we apply the associative property that we can move the parentheses in an expression to help us with our work.
Let’s look at an example of a numerical expression.
Example
\begin{align*}\left(\frac{1}{3}+\frac{1}{4}\right)+\frac{2}{4}\end{align*}
Here we have an expression that has three rational numbers in it. We can see that there are two different denominators. However, look at the groupings. If we move the parentheses to group the fourths together, it will help us with the addition.
This is an example of the Associative Property of Addition.
\begin{align*}\frac{1}{3}+\left(\frac{1}{4}+\frac{2}{4}\right)\end{align*}
Now we can find a solution.
\begin{align*}\frac{1}{3}+\frac{3}{4}=\frac{4}{12}+\frac{9}{12}=\frac{13}{12}=1 \frac{1}{12}\end{align*}
Now let’s look at an example of a variable expression where the associative property would be useful.
Example
\begin{align*}\left(x+\frac{4}{5}\right)-\frac{2}{5}\end{align*}, when \begin{align*}x\end{align*} is \begin{align*}\frac{3}{7}\end{align*}
First, we can substitute three-sevenths into the expression for the variable.
\begin{align*}\left(\frac{3}{7}+\frac{4}{5}\right)-\frac{2}{5}\end{align*}
Next, you can see that it makes much more sense to work with the fifths and then the sevenths. Let’s regroup these fractions and find a solution.
\begin{align*}& \frac{3}{7}+\left(\frac{4}{5}-\frac{2}{5}\right)\\ & \frac{3}{7}+\frac{2}{5}=\frac{15}{35}+\frac{14}{35}=\frac{19}{35}\end{align*}
II. Identify and Apply the Commutative Property of Addition and Multiplication in Rational Number Operations, using Numerical and Variable Expressions
We can also apply the Commutative Property when we evaluate expressions with rational numbers in them. Rearranging the numbers we are working with can help us to simplify our efforts.
Let’s look at an example.
Example
\begin{align*}.56+\frac{1}{2}+.24\end{align*}
In this example, we have three rational numbers that we are adding. If you look, it doesn’t make sense to add the decimal and the fraction and then the decimal. It makes more sense to use the commutative property to rearrange the values. We add the decimals first and then deal with the fraction.
\begin{align*}.56+.24.+\frac{1}{2}\end{align*}
Now we can add the decimals together.
\begin{align*}& .56 + .24 = .80\\ & .80+\frac{1}{2}\end{align*}
Here we are trying to add a decimal and a fraction. We need to convert the fraction to a decimal so that the rational numbers are in the same form. Then we can easily add them.
\begin{align*}\frac{1}{2} &= .50\\ .80+.50 &= 1.3\end{align*}
Now let’s look at an example with a variable expression and multiplication.
Example
\begin{align*}\frac{1}{2} \cdot (.56) \cdot \frac{1}{3}\end{align*}
Here is a numerical expression with two fractions and a decimal. Notice that the decimal is in the middle. It makes the most sense to rearrange the fractions so that you can multiply them together, and then multiply the decimal.
\begin{align*}\frac{1}{2} \cdot \frac{1}{3}=\frac{1}{6}\end{align*}
Now let’s convert .5 to a fraction and multiply it by one-sixth.
\begin{align*}\frac{1}{6} \cdot \frac{1}{2}=\frac{1}{12}\end{align*}
III. Identify and Apply the Distributive Property in Rational Number Operations, using Numerical and Variable Expressions
We can use the Distributive Property to help us simplify expressions with parentheses and operations within the parentheses. Let’s look at an example.
Example
\begin{align*}\frac{1}{2} \left(\frac{3}{6}+8\right)\end{align*}
In this example, we need to multiply the term outside of the parentheses with both of the terms inside the parentheses. Then we can simplify the expression further.
\begin{align*}\frac{1}{2} \cdot \frac{1}{2} &= \frac{1}{4}\\ \frac{1}{2} \cdot 8 &= 4\\ 4+\frac{1}{4} &= 4 \frac{1}{4}\end{align*}
We can also use the Distributive Property to simplify variable expressions.
Example
\begin{align*}x(3+7)\end{align*}
Now we can distribute the variable to both of the terms in the parentheses and then we can simplify the expression.
\begin{align*}& 3x+7x\\ & 10x\end{align*}
IV. Model and Solve Real – World Problems Using Equations Involving Rational Numbers
Let’s look at using what we have learned to solve some real-world problems with rational numbers.
Example
## Time to Practice
Directions: Use what you have learned to work with each problem concerning rational numbers.
1. Write an equation that shows the commutative property of addition.
2. Write an equation that shows the zero property of multiplication.
3. What is the difference between the additive identity and the multiplicative identity properties?
4. The following shows a student’s work: \begin{align*}\left(\frac{3}{3}\right) \left(\frac{1}{5}\right)+ \left(\frac{5}{5}\right) \left(\frac{2}{3}\right)=\frac{3}{15}+\frac{10}{15}\end{align*}. What property did the student use to find a common denominator?
5. The expression \begin{align*}2p^3-4m\end{align*} can be used to find the sales profit of a company where \begin{align*}p\end{align*} is the number of products they sell and \begin{align*}m\end{align*} is the number of miles they travel. If they sold 5 products and traveled 10 miles, what was their profit?
6. The expression \begin{align*}\frac{11s}{2}+7t\end{align*} can be used to find the group admission price, where \begin{align*}s\end{align*} is the number of students and \begin{align*}t\end{align*} is the number of teachers. If there are 20 students and 4 teachers, what is the group admission price?
7. Brooke needs to save $146 for a trip. She has$35 in her savings account. She saves $15.75 each week. She also has to spend$15 to buy a present for a friend. How many weeks will Brooke need to save to have enough for her trip?
8. Vinnie is \begin{align*}\frac{1}{2}\end{align*} as old as Julie. In 6 years, Julie will be 24. How old is Vinnie?
9. Manuel starts with $30. He earns$8.50 per hour plus an additional bonus of $12 each day. He spends$7.50 for lunch. If he has \$94 at the end of the day, for how many hours did he work?
10. A formula for the perimeter of a rectangle is \begin{align*}P=2(l+w)\end{align*}, where \begin{align*}P\end{align*} is the perimeter, \begin{align*}l\end{align*} is the length, and \begin{align*}w\end{align*} is the width. If the perimeter of a rectangle is 312 centimeters and the width is 67.3 centimeters, what is the length?
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Sampling and digitization of time series
There is a time series of 100 data points(say). I wish to assign symbols of 0 1 2 for each unique data point. The issue is I have tried but got stuck since no matter I specify the symbols,the program just outpits probability of 1's and 0's. The following is the issue
1. The statement
s=x(:,1) > 0.5;
outputs a binary result 0,1 . So,how do I create multiple partitions / discretization so that apart from 0,1 other numerals can also be assigned.
Is there any other way to symbolize and partition?
-
Have a look at the `histc` function. – Ben Voigt Jul 16 '13 at 1:55
The obvious way to do this would be something like:
``````s=zeroes(size(x,1), 1);
s(x>=BP(1) & x<BP(2)) = 1;
s(x>=BP(2)) = 2;
etc.
``````
where BP is your list of break points (i.e., the edges of the partitions). That would make everything below BP(1)=0, things between BP(1) and BP(2) =1, and entries above BP(2) = 2;
I imagine something like this ought to work too:
``````s = zeroes(size(x,1), 1];
for ii=1:length(BP)
idx = x > BP(ii);
s(idx) = s(idx) + 1;
end
``````
You've got more options if there are some constraints on your data and/or bin size. You might consider some clever combination of multiplication, division and rounding/truncating. For example, suppose your data was all in the range [0, 1) and you wanted it divided into twenty evenly spaced bins. Then, you could do something like:
``````s = floor(x(:,1) .* 20);
``````
which would make s take values between 0 and 19. If your data wasn't already in that interval, you could obviously rescale it first:
``````data = x(:,1);
data = data - min(data);
data = data ./ (max(data) + eps(max(data)));
s = floor(data .* 20);
``````
Note that here, the normalizing factor in line 3 is not max(data), but the next largest number that matlab can represent. We do that so that there are 20 groups and not 21.
-
Thank you for your reply.I am unable to understand the meaning of breakpoints.Are they coordinates OR any arbitrary number depending on my data set (BP(1)=0.5 say,BP(2)=1.5 etc). – Chaitali Jul 12 '12 at 1:21
Further, I quite did not follow the x,y concept.Could you be kind enough to elaborate on that also,what it means and how to determine the BP's. – Chaitali Jul 12 '12 at 1:35
By breakpoints, I meant the edges of your partitions; they could be whatever number you choose. For the second, example, I should have stuck with the conventions in your post. I'll try editing it to make it clearer. Let me know if that helps. – Matt Krause Jul 12 '12 at 4:00
Thank you for the clarification.However, some doubts still persist. (A)Is there a way to determine the breakpoints in general without looking at the data set? (B)What if I want to assign a range of points say BP1=0.5 then all data points in range 0-0.5 fall under BP1; BP2=1 then all data points in range 0.5-1 classify under BP2 and so on. – Chaitali Jul 12 '12 at 4:54
That's exactly what the first snippets do. You'd do something like BP=[0.5, 1]. They don't come from the data at all--you've got to provide them. The only trick here is what matlab calls "logical indexing". That is, instead of providing a numeric index (i.e., a(4) gets you the 4th element of the array), you can also provide a list of ones/zeros (i.e., true and falses), so a(logical(0,0,0,1)) also gets you the 4th element. This works as both and lvalue and an rvalue, so you can use them to get or set variables. – Matt Krause Jul 12 '12 at 20:46
show 1 more comment
The statement `x(:,1) > 0.5;` is creating a logical index: `true (1)` where the condition is satisfied, `false (0)` where it is not. You can use this logical index to grab values from the original vector where the condition is satisfied.
``````logical_index = x(:,1) > 0.5;
s = x(logical_index,1); %# select the subset of the matrix given by x > 0.5
``````
`s` now contains the values from `x` that are greater than 0.5.
Beyond this, I can't understand what you're trying to do. An small example data set would help (if you still need help, that is).
Edit: To find values appropriate for dividing your set up this way, take a look at prctile. You can then apply any of the methods in the answers to figure out which elements fall into which category.
-
Thank you for your prompt reply.As mentioned in Answer1,i intend to create partion in the data which spans say there are 10 points in range 0.5-1(so data points falling in this range will be denoted by symbol 1);another 10 ponts in 1-1.5(denoted by symbol 2) and so one.So,without looking at the data points is there way to describe the partitions so that they are valid for all cases? – Chaitali Jul 12 '12 at 2:16
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Test: Progression (AP And GP)- 2
# Test: Progression (AP And GP)- 2
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## 10 Questions MCQ Test CSAT Preparation for UPSC CSE | Test: Progression (AP And GP)- 2
Test: Progression (AP And GP)- 2 for Banking Exams 2022 is part of CSAT Preparation for UPSC CSE preparation. The Test: Progression (AP And GP)- 2 questions and answers have been prepared according to the Banking Exams exam syllabus.The Test: Progression (AP And GP)- 2 MCQs are made for Banking Exams 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Progression (AP And GP)- 2 below.
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Test: Progression (AP And GP)- 2 - Question 1
### Find the 15th term of an arithmetic progression whose first term is 2 and the common difference is 3
Detailed Solution for Test: Progression (AP And GP)- 2 - Question 1
Test: Progression (AP And GP)- 2 - Question 2
### What is the sum of the first 15 terms of an A.P whose 11 th and 7 th terms are 5.25 and 3.25 respectively
Detailed Solution for Test: Progression (AP And GP)- 2 - Question 2
Test: Progression (AP And GP)- 2 - Question 3
### If(12+22+32+…..+102)=385,then the value of (22+42+62 + …+202) is :
Detailed Solution for Test: Progression (AP And GP)- 2 - Question 3
(12+22+32+.......102)=385
(22+42+62+.......+202) = 22(12+22+ 32+.....+102 )
=4(385)
=1540
Test: Progression (AP And GP)- 2 - Question 4
In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?
Detailed Solution for Test: Progression (AP And GP)- 2 - Question 4
Test: Progression (AP And GP)- 2 - Question 5
The sum of the first 100 numbers, 1 to 100 is divisible by
Detailed Solution for Test: Progression (AP And GP)- 2 - Question 5
Test: Progression (AP And GP)- 2 - Question 6
How many terms are there in G.P 3,6,12,24,….,384?
Detailed Solution for Test: Progression (AP And GP)- 2 - Question 6
Test: Progression (AP And GP)- 2 - Question 7
Four angles of a quadrilateral are in G.P. Whose common ratio is an intiger. Two of the angles are acute while the other two are obtuse. The measure of the smallest angle of the quadrilateral is
Detailed Solution for Test: Progression (AP And GP)- 2 - Question 7
Test: Progression (AP And GP)- 2 - Question 8
How many numbers between 11 and 90 divisible by 7?
Detailed Solution for Test: Progression (AP And GP)- 2 - Question 8
Test: Progression (AP And GP)- 2 - Question 9
If Sn denotes the sum of the first n terms in an Arithmetic Progression and S1: S4 = 1: 10
Then the ratio of first term to fourth term is:
Detailed Solution for Test: Progression (AP And GP)- 2 - Question 9
Test: Progression (AP And GP)- 2 - Question 10
The sum of the three numbers in A.P is 21 and the product of their extremes is 45. Find the numbers.
Detailed Solution for Test: Progression (AP And GP)- 2 - Question 10
Let the numbers are be a - d, a, a + d
Then a - d + a + a + d = 21
3a = 21
a = 7
and (a - d)(a + d) = 45
a2 - d2 = 45
d2 = 4
d = +2
Hence, the numbers are 5, 7 and 9 when d = 2 and 9, 7 and 5 when d = -2. In both the cases numbers are the same.
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# Canonical rotations that do not produce computational singularities
Intro
On the topic of dynamical systems associated with 3-dimensional rotation of rigid bodies, you will always encounter singularities in the equations of motion that will produce computational errors at certain rotations. At least, these singularities will always occur if the following approach is attempted (see next section).
Problem
Suppose you want to describe the coordinates $$(x,y,z)$$ of a rigid body by spherical parameterization using two canonical coordinates $$q_1$$ and $$q_2$$ as spherical coordinates
$$\vec r = \left( {\begin{array}{*{20}{c}} {r\sin q_1 \cos q_2 }\\ {r\sin q_1 \sin q_2 }\\ {r\cos q_1 } \end{array}} \right).$$
You can easily obtain the velocity vector $$\vec {\dot r}$$ and then associate it with the kinetic energy $$T = \frac{1}{2}m{\left| {\vec {\dot r}} \right|^2}.$$ To further simplify the problem, assume the potential energy to be 0. The Lagrangian $$L = T - V$$ and Hamiltonian $$H = T + V$$ can then be obtained.
From the Lagrangian we can obtain the equations of motion $$\begin{array}{l} {{\ddot q}_1} = \frac{1}{2}\dot q_2^2\sin \left( {2{q_1}} \right)\\ {{\ddot q}_2} = 2{{\dot q}_1}{{\dot q}_2}\sin \left( {2{q_1}} \right)\frac{1}{{\cos \left( {2{q_1}} \right) - 1}}. \end{array}$$ It is clear that the equations of motion are ill-defined (in a computational sense) for $$q_1 = n\pi$$, which is at the poles of the sphere.
Let us examine the Hamiltonian equations to verify that the issue persists. Using canonical momenta $$p_1$$ and $$p_2$$, we find:
$$\begin{array}{l} {p_1} = {{\dot q}_1}m{r^2}\\ {p_2} = - \frac{1}{2}{{\dot q}_2}m{r^2}\left( {\cos \left( {2{q_1}} \right) - 1} \right)\\ H = - \left( {p_1^2 - p_1^2\cos \left( {2{q_1}} \right) + 2p_2^2} \right)\frac{1}{{2m{r^2}( \cos \left( {2{q_1} } \right) - 1)}}\\ \frac{{\partial H}}{{d{q_1}}} = - 2p_2^2\sin \left( {2{q_1}} \right)\frac{1}{{m{r^2}{{\left( {\cos \left( {2{q_1}} \right)} -1 \right)}^2}}}\\ \frac{{\partial H}}{{d{q_2}}} = 0\\ \frac{{\partial H}}{{d{p_1}}} = \frac{{{p_1}}}{{m{r^2}}}\\ \frac{{\partial H}}{{d{p_2}}} = - \frac{{2{p_2}}}{{m{r^2}\left( {\cos \left( {2{q_1}} \right) - 1} \right)}}. \end{array}$$
The problem persists: The equations of motions are always ill-defined (from a computational point of view) at the poles. Note that the limit is never diverging and the solution makes perfect sense in a physical point of view. From a computational point of view, any numerical simulation will produce large numerical errors if the rotation $$q_1$$ approaches $$n\pi$$. This would be a problem if the system was a spherical pendulum for instance.
The same issue occurs using cylindrical coordinates. Using other alternative representations of $$\vec r$$ does not appear to solve the issue either.
Question
Is it possible to describe the equations of motion for rigid body rotation without having to deal with the threat of singularities?
I am aware you can use conditional coordinate transformations during simulation to address the issue. This poses some other issues for multistep integrators, and I therefore do not desire this option.
How would you approach the problem of describing the equations for rotating rigid bodies without computational singularities, specifically preventing that the Hamiltonian gradient and the canonical coordinates approaches infinity at any point?
• Perhaps using a different parameterization of the canonical coordinates?
• Perhaps using more than two canonical coordinates to describe the system?
• Perhaps making the system time-dependent?
• Perhaps taking advantage of the cyclic variable - in this case $$q_2$$ with $$\frac{{\partial H}}{{d{q_2}}} = 0$$ to somehow reduce the system
• Perhaps some transformations of coordinates
• Perhaps using quaternions (suggested by G. Smith, JEB and suggested transformation by Eli). How would you describe the equations of motions with quaternions without introducing singularities?
Open for any ideas
To further specify what I seek: A set of generalized coordinates from which the equations of motion associated with the rotation can be expressed without intrusive divergence or singularities.
I pursue suggestions that can be utilized also for more complicated rigid body systems with non-zero potential energy. For instance the double spherical pendulum and other systems with cyclic properties.
Progress
Quaternions coordinates were tested. The Hamiltonian and Lagrangian equations of motion can be expressed in terms of three canonical/generalized quaternion coordinates. Unfortunately divergence occurs $$a^2 + b^2 + c^2 = 1$$ due to division of the fourth coordinate $$d = 0$$. Divergence occurs more often than systems of two canonical coordinates, and this approach is therefore not pursued any further.
2) epsilon to remove the singularity (proposed by Eli)
A small value $$\epsilon$$ can be added to the divisor in either the Hamiltonian or Lagrangian equations of motion, preventing division by 0. This can be done using both spherical and cylindrical coordinates.
Unfortunately this method changes the system noticeable when $$\epsilon$$ is large. If $$\epsilon$$ is small, an error similar to the effect of the singularity will be produced. This solution is very practical, but not ideal for accurate depictions of the system.
3) Variable axis of singularity
We can construct the coordinate system of the spherical pendulum (or other cyclic systems) using two pairs of coordinate $$\vec r_x$$ and $$\vec r_y$$. When the x-coordinate approaches its maximum and minimum value, a singularity is appraoched for coordinate system $$\vec r_x$$. Likewise for the y-coordinate. The idea is then to transform the coordinates from the $$\vec r_x$$ to $$\vec r_y$$ whenever the x-singularity is approached, and transform form $$\vec r_y$$ to $$\vec r_x$$ whenever the y-singularity is approached.
$${\vec r_x} = \left( {\begin{array}{*{20}{c}} {r\cos \left( {{q_1}} \right)}\\ {r\sin \left( {{q_1}} \right)\sin \left( {{q_2}} \right)}\\ {r\sin \left( {{q_1}} \right)\cos \left( {{q_2}} \right)} \end{array}} \right),{\vec r_y} = \left( {\begin{array}{*{20}{c}} {r\sin \left( {{q_1}} \right)\sin \left( {{q_2}} \right)}\\ {r\cos \left( {{q_1}} \right)}\\ {r\sin \left( {{q_1}} \right)\cos \left( {{q_2}} \right)} \end{array}} \right)$$
You can transform coordinates from one system to the other using $$\left( {\begin{array}{*{20}{c}} {{q_1}}\\ {{q_2}} \end{array}} \right) \to \left( {\begin{array}{*{20}{c}} {{{\cos }^{ - 1}}\left( {\sin \left( {{q_1}} \right)\sin \left( {{q_2}} \right)} \right)}\\ {{\rm{atan2}}\left( {cos\left( {{q_1}} \right),\sin \left( {{q_1}} \right)\cos \left( {{q_2}} \right)} \right)} \end{array}} \right)$$
https://youtu.be/5bh_dMn-Plc
Above video above illustrates this method, where the simulation switches between $$\vec r_x$$ (red sphere) and $$\vec r_y$$ (blue sphere). The simulation never reaches singularities, but the transformation between the two systems is not symplectic and produces a linear energy dissipation (energy error depicted top-right in the video). The transformation needs to be done in a different computational way. Investigating this approach further in hope of a proper solution.
4) Other ideas?
If you have another idea - either a new approach or one based on the above, please share it. This post will be updated when further progress is made.
• Aren’t quaternions good for avoiding such problems? Commented Jul 27, 2020 at 5:01
• Would Computational Science be a better home for this question? Commented Jul 27, 2020 at 7:07
• If I am to follow G. Smith's and JEBs suggestion, the challenge is to express the Hamiltonian or Lagrangian with quaternion canonical coordinates. So it feels more like a physics problem to me. But this problem apparently evaded all physics textbooks and internet resources on rigid body systems...
– user270876
Commented Jul 27, 2020 at 7:17
• Read page 7 (above) for references to a discussion on rotational representations with and without singularities. Commented Aug 1, 2020 at 21:00
• malcolmdshuster.com/Pub_1993c_J_RotVec_IEEE.pdf is a paper that I think solves this problem, or at least provides useful insight into past efforts. Commented Aug 1, 2020 at 21:25
It's not just pendulums. It's a major issue for spacecraft navigation, especially if, for instance, your trajectory involves the deployment of a supersonic parachute on Mars.
Hence: quaternions are the standard.
They are also computationally faster, which can be a factor with space-qualified CPUs, which are not fast.
• Answer to both JEB and G. Smith. You are correct that quaternions are used to describe the rotations without nasty singularities and gimbal lock. However, I am yet to see Hamiltonian systems with non-zero potential being expressed in terms of quaternion coordinates. Do you have an idea how to describe the kinetic energy and potential energy (such as the spherical pendulum with $V = mgz$) in terms of quaternion canonical coordinates?
– user270876
Commented Jul 27, 2020 at 5:25
I don't think you can avoid this singularity by using Quaternions because they are good for rotation .
but for numerical simulation you can use this concept:
$$\ddot{\theta}=-2\,{\frac {\dot{\theta}\dot{\phi}\cos \left( \phi \right) }{\sin \left( \phi \right) }} \tag 1$$
and
$$\ddot{\phi}={\frac {\sin \left( \phi \right) \left( r{\dot \theta }^{2}\cos \left( \phi \right) +g \right) }{r}}\tag 2$$
you have singularity if you want to simulate equation (1) ans (2) with initial condition $$\phi(0)=0$$ , to avoid this singularity substitute in equation (1) $$\sin(\phi)\mapsto \sin(\phi+\epsilon)$$ where $$\epsilon$$ is a small number
$$\ddot{\theta}=-2\,{\frac {\dot{\theta}\dot{\phi}\cos \left( \phi \right) }{\sin \left( \phi+\epsilon \right) }} \tag 3$$
if the initial condition $$\dot{\theta}(0)$$ equal zero or $$\dot{\phi}(0)$$ equal zero you don't get problem with the initial condition $$\phi(0)=0$$
This is the simulation result ,geodetic line ,with initial conditions all zero and gravitation g, I choose $$\epsilon=0.001$$
edit:
Sphere Position Vector:
$$\vec{R}=\left[ \begin {array}{c} r\cos \left( \theta \right) \sin \left( \phi \right) \\ r\sin \left( \theta \right) \sin \left( \phi \right) \\ r\cos \left( \phi \right) \end {array} \right] \tag 1$$
where $$\theta$$ the azimuth coordinate $$0\le\theta\le 2\pi$$ , $$\phi$$ the polar coordinate $$0\le \phi\le \pi$$ and r is the radius of the sphere .
How to avoid the singularity at $$\phi=0$$ and $$\phi=\pi$$
substitute in equation (1) $$\sin(\phi)\mapsto \sin(\phi+\epsilon)~,$$ ( with r=1 )
the kinetic energy is now
$$T=1/2\, \left( \left( \cos \left( \phi+\epsilon \right) \right) ^{2}+1 - \left( \cos \left( \phi \right) \right) ^{2} \right) m{\dot\phi }^{2}+ 1/2\,m \left( 1- \left( \cos \left( \phi+\epsilon \right) \right) ^{2 } \right) {\dot\theta }^{2}$$
to inspect the singularity you obtain the Mass Matrix from the kinetic energy $$M(\theta \,,\phi )=\left[ \begin {array}{cc} {\frac {\partial ^{2}}{\partial {\theta p}^ {2}}}T \left( \theta p,\phi p \right) &{\frac {\partial ^{2}}{ \partial \theta p\partial \phi p}}T \left( \theta p,\phi p \right) \\ {\frac {\partial ^{2}}{\partial \theta p\partial \phi p}}T \left( \theta p,\phi p \right) &{\frac {\partial ^{2}}{ \partial {\phi p}^{2}}}T \left( \theta p,\phi p \right) \end {array} \right]$$
where $$\theta p=\dot{\theta}$$ and $$\phi p=\dot{\phi}$$
$$M=\left[ \begin {array}{cc} -m \left( -1+ \left( \cos \left( \phi+ \epsilon \right) \right) ^{2} \right) &0\\ 0& \left( \left( \cos \left( \phi+\epsilon \right) \right) ^{2}+1- \left( \cos \left( \phi \right) \right) ^{2} \right) m\end {array} \right]$$ thus the determinate is: $$\det(M)=-{m}^{2} \left( -1+ \left( \cos \left( \phi+\epsilon \right) \right) ^{2} \right) \left( \left( \cos \left( \phi+\epsilon \right) \right) ^{2}+1- \left( \cos \left( \phi \right) \right) ^{2} \right)$$ take the determinate equal zero and solve for $$\phi$$ you obtain two real solutions
$$\phi_0=-\epsilon ~,\phi_0=-\epsilon+\pi$$
thus for $$\epsilon\ll$$ the simulation will work perfect
• Your first suggestion on modifying the equations of motion with eps will not do the trick, as my concern is not the invalid initial condition, but rather the spurious Hamiltonian dissipation that arise when the singularity is approached.
– user270876
Commented Jul 27, 2020 at 19:09
• @Egeris The Transformation Matrix is correct but the equation of motion is not, because d is also a function of the time, so it is a little bit complicate, I can give you the answer for that but I need your quotation "How to implement the Quanternios "
– Eli
Commented Jul 28, 2020 at 20:06
• Typo: It is Quaternions not Quaternios. Commented Jul 29, 2020 at 12:03
• I obtained the correct equations from the quaternion rotation matrix though. But it isn't useful unless I can make it work with transformations during the simulation to skip the singularities. Currently having some luck doing simulations transforming between two coordinate pairs of different spherical coordinates.
– user270876
Commented Jul 30, 2020 at 8:32
• @Eli It seems like a practical approach for approximating the time evolution for simple spherical systems. The challenge for me, is the to obtain $exactly$ the phase space from the time evolution, but I am yet to find a method that achieves this. I'll do a comparative analysis of this approach against my other techniques involving moving axes, once I manage to achieve symplecticity.
– user270876
Commented Aug 3, 2020 at 13:16
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# What are the best strategies for identifying where to begin unit conversions?
Dec 4, 2014
To begin unit conversions you want to write down all the given information you have in the problem including what the question is asking for.
Example: How many moles of sodium are there in a 0.59 gram sample of sodium metal.
1. Given info: 0.59 grams Na
moles Na=?
In this simple example we see that the question is asking for the moles of sodium, so when we know that when we convert we should be left with moles as our only unit .
2. Now that you know what units you need to end up with, we need to find something that relates our given units ( grams ) to our desired unit ( moles .)
A simple connection for grams$\to$moles is given by the molar mass !
1 mol Na=23 grams Na
Now , Set up your problem:
0.59 grams Na $\cdot \frac{1 m o l N a}{23 g r a m s N a} = 0.026 m o l N a$
Since we cancelled our units properlly ($g r a m s \cdot \frac{m o l}{g r a m s} = m o l$) we know that our solution must be correct!
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# What is the number of possible $x$ values in $\frac{x}{100}= \sin(x)$ [duplicate]
Problem $\frac{x}{100}= \sin(x)$ We are asked to find the number of possible values of $x$ in this scenario and I had tried to figure it out by the use of trigonometric identities but then i had realized that there are no trig identities that can help me... or is there?
steps that i had tried: I first multiplied 100 on both sides to get $x$ by itself to get: $100(\sin x)=x$
Then i determined that $\sin x \le 1$ therefore I had determined that $\frac{x}{100}$ is $\le 1$
But when i look at the answer choices all of them are less than $100$ meaning that all of them are going to make $\frac{x}{100}$ less than 100 when $x$ is substituted.
## marked as duplicate by user99914, user228113, Em., Community♦Jul 12 '16 at 6:08
• @angelo mark I understand that you are a perfectionist but maybe changing all the x's into $x$ is a bit too much? – John Rawls Jul 11 '16 at 7:25
• Sorry. I'll remember your username for not editing in future. Really Sorry. – Angelo Mark Jul 11 '16 at 7:55
• This question has already been answered. See here. – Bernard Jul 11 '16 at 8:12
• @AngeloMark don't worry man, i do the same and win 2 points. Lol – julio godoy Jul 11 '16 at 9:20
Since $|\sin x|\le 1$, one needs only to consider $\frac{|x|}{100} \le 1$, or $|x|\le 100$. Since both functions are odd, we restrict ourselves to the interval $[0,100]$. The curve
$$\tag{1} y = \frac{ x}{100}$$
is positive on $(0,100]$. $\sin x$ is periodic with period $2\pi$. On each (half) period
$$[0,\pi], [2\pi, 3\pi], \cdots [2n\pi, (2n+1)\pi]$$
as long as $(2n+1) \pi <100$, the curve $(1)$ intersects $\sin x$ at $2$ points. Since $15.5< 100/2\pi <16$, this means that there are $32$ nonnegative solution and thus $63$ solution in the real line.
• Are you did not make any errors? because the solution choices are 61, 62, 63. 64, or 65 – John Rawls Jul 11 '16 at 7:45
• @JohnRawls Yes, I suck at counting. – user99914 Jul 11 '16 at 7:48
• i'm sure you are right but can u explain on how u set 15<100/2π<16 and why this means that there are 32 nonnegative solutions and 63 real solutions? – John Rawls Jul 11 '16 at 7:51
• @JohnRawls : $15<100/2\pi<16$ are obtained by direct calculation (indeed I need $15.5 <100/2\pi$. This implies that there are $16$ such $[2n\pi, (2n+1)\pi]$ in $[0,100]$ and each gives me $2$ solutions. Thus we have $32$ in $[0,100]$. There are also $32$ solutions in $[-100,0]$, but that $0$ is in both counting, so the total number should be $32\times 2 -1 = 63$. – user99914 Jul 11 '16 at 7:55
• Really, I just plug in $100/2\pi$ in wolfram alpha and got around 15.9. – user99914 Jul 11 '16 at 7:58
look at this site body must be [1]: https://www.wolframalpha.com/input/?i=x%2F100%20%3D%20sin(x)
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23 Jan 2021
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Learn about triangle calculation to find H, triangle area calculator and right angled triangle calculation. b = 6. c = 7. This example will show you how to read user inputs in C++ and how to do mathematical calculations.. Before moving to the program, let me quickly show you the mathematical formula to calculate the triangle area. And we are good at identifying simplicity. In case you need them, here are the Trig Triangle Formula Tables, the Triangle Angle Calculator is also available for angle only calculations. Practice Questions on Triangle Calculation. Status: To find the third side "c" Pythogoras Theorem will be used which is given by the following formula. $$c^2 = 576 + 49 = 625$$. It can be treated as two right triangles and the Pythagorean Theorem can be applied on them to find the third sides. We will take the sides as input from the user. In case of right angled triangle, the side "c" (or hypotenuse) is calculated using the Pythagorean Theorem. Math Warehouse's popular online triangle calculator: Enter any valid combination of sides/angles(3 sides, 2 sides and an angle or 2 angle and a 1 side) , and our calculator will do the rest! a=7 β=40 mc=5... triangle calc by one side, one angle, and one median. Triangle Calculator Instructions. Preface. Now, you will be able to easily solve problems on triangle calculation to find H, triangle area calculator and right angled triangle calculation. Isabella took an example of a right-angled triangle. check out the law of sines ambiguous case. The number of significant values entered will determine the number of significant figures in the results. 3 2 + 4 2 = 5 2. $$\therefore$$ Michelle found length of third side is 12.02 units. The interior angles of a triangle always add up to 180° while the exterior angles of a triangle are equal to the sum of the two interior angles that are not adjacent to it. There are four types of isosceles triangles: acute, obtuse, equilateral, and right. A Pythagorean triple is a set of 3 positive integers for sides a and b and hypotenuse c that satisfy the Pythagorean Theorem formula a2 + b2 = c2. Use the calculator to calculate coordinates of the centroid of the triangle ABC.Enter the x,y coordinates of each vertex, in any order. The Law of Cosines is given by the following formula: It is given as the ratio of subtraction of length of opposite side squared from the sum of the lengths of adjacted sides squared to the product of two times the lengths of adjacent sides. Moreover it allows specifying angles either in grades or radians for a more flexibility. b = c * sin(β) or b = c * cos(α) Given angle and one leg; Find the missing leg using trigonometric functions: a = b * tan(α) b = a * tan(β) Given area and one leg; As we remember from basic triangle area formula, we can calculate the area by multiplying triangle height and base and dividing the result by two. Angle 3 and Angle C fields are NOT user modifiable. Explore triangle calculation:find c with our Math Experts in Cuemath’s LIVE, personalized and interactive online classes. This calculator calculates for the length of one side of a right triangle given the length of the other two sides. It is not possible for a triangle to have more than one vertex with internal angle greater than or equal to 90°, or it would no longer be a triangle. Use a calculator to estimate the square root to one decimal place. There are 4 common rules for solving a triangle, as explained below. Enter values three of the six sides and angles of the triangle and the other three values will be computed. 9 + 16 = 25. A right triangle is a kind of triangle that has one angle that measures C=90°. "C" generally represents the third side in a triangle. Again, this right triangle calculator works when you fill in 2 fields in the triangle angles, or the triangle sides. To find the area of triangle having sides A,B and C, we use Heron's formula. Hence, the final equation obtained is $$a^2+b^2 = c^2$$. Right Triangle Calculator. 25 = 25. Given sides a and b find side c and the perimeter, semiperimeter, area and altitudes. Let's learn how to calculate value of c in a triangle. 3. The algorithm of this right triangle calculator uses the Pythagorean theorem to calculate the hypotenuse or one of the other two sides, as well as the Heron formula to find the area, and the standard triangle perimeter formula as described below. In a Right triangle, the side c that is opposite of the C=90° angle, is the longest side of the triangle and is called the hypotenuse. Help her visualize the triangle and do the triangle calculation to find C. Michelle will draw the triangle as follows: To find the third side, Michelle uses the Law of Sines which is given as follows: Given the value of sides a and b, 6 units and 10 units respectively and A = 30∘. Their angles are also typically referred to using the capitalized letter corresponding to the side length: angle A for side a, angle B for side b, and angle C (for a right triangle this will be 90°) for side c, as shown below. Home / Mathematics / Trigonometric functions (Deg) Calculates the three angles and area of a triangle given three sides. This final equation is the formula of Pythagorean Theorem. Select which side of the right triangle you wish to solve for (Hypotenuse c, Leg a, or Leg b). Point in triangle. You can find the hypotenuse: Given two right triangle legs; Use the Pythagorean theorem to calculate the hypotenuse from right triangle sides. Step #3: Enter the two known lengths of the right triangle. Altitude c of Right Triangle: h c = (a * b) / c. 1. in Quadrant II, for more information on this topic, In these cases, in actuality , the calculator is really producing correct The trigonometric ratios used to find angles A and B are given by sin(A) = a / h , A = arctan(a / h) sin(B) = b / h , B = arctan(b / h) The area and perimeter of the right triangle are given by Area = (1/2) a b . Learn about triangle calculation to find H, triangle area calculator and right angled triangle calculation in the concept of triangle calculation to find C. Check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.. Help Isabella do the triangle calculation to find H. Isabella knows area of triangle = \dfrac{1}{2} \times b \times a. Here you can enter two known sides or angles and calculate unknown side ,angle or area. An acute isosceles triangle is a triangle with a vertex angle less than 90°, but not equal to 60°.. An obtuse isosceles triangle is a triangle with a vertex angle greater than 90°.. An equilateral isosceles triangle is a triangle with a vertex angle equal to 60°. This is the acute triangle How will Leonard do the triangle calculation to find hypotenuse of a right angled triangle if the area of triangle is 84 unit2 and the perpendicular is 24 units? Area of triangle as per the above figure will be given by formula. Upon making your selection the triangle calculator will load the appropriate entry form. Circumcenter calculator is used to calculate the circumcenter of a triangle by taking coordinate values for each line. and experience Cuemath’s LIVE online class with your child. ma=1 mb=2.5 mc=2... triangle calc by three medians. The calculator below determines if a given point is inside a 2D triangle. The calculator uses a simple algorithm based on vector cross product features. We can divide the triangle into two halves. Learn about triangle calculation to find H, triangle area calculator and right angled triangle calculation in the concept of triangle calculation to find C. Check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page. The smallest known Pythagorean triple is 3, 4, and 5. 1. You may adjust the accuracy of your results. side a: side b: side c: angle A ° = angle B ° = angle C ° = height h . 2. Area of a triangle in C++ : In this C++ program, we will learn how to find the area of a triangle if its sides are given. ASA - a side and 2 adjacent angles. The tool which is used to find the long side of the right triangle is the hypotenuse calculator. Male Female Age Under 20 years old 20 years old level 30 years old level 40 years old level 50 years old level 60 years old level or over Occupation Elementary school/ Junior high-school student Here it means the size. In case of a right-angled triangle, it is referred to as "c" or the hypotenuse (h). Select/Type your answer and click the "Check Answer" button to see the result. Angles of a triangle Calculator . Enter any valid input (3 side lengths, 2 sides and an angle or 2 angle and a 1 side) and our calculator will do the rest. On applying the Law of Sines she gets, $$\dfrac{8}{\sin 30}$$= $$\dfrac{12}{\sin B}$$, Hence, $$\sin B$$ = $$\dfrac{12 \times \sin 30}{8}$$. Help her do the triangle calculation to find C and angles A and C. To find the third side, Claire uses the Law of Sines which is given as follows: $$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$$. Math represents ideas, creative-thinking, and problem-solving. Vertex A. Vertex B. Vertex C. Point P. Calculation precision. However, it is then rounding them for you- which leads to seemingly inaccurate results and possible Sides "a" and "b" are the perpendicular sides and side "c" is the hypothenuse. SSS - 3 side lengths. Michelle's teacher asked her to find the third side of triangle, if the length of a and b was mentioned as 6 units and 10 units respectively and the value of angle A was 30∘. It will even tell you if more than 1 triangle can be created. We can find the third side of a triangle using any of the laws, either the Law of Sines or Law of Cosines can be used. Make your kid a Math Expert. The formula for the area of a triangle is side x height, as shown in the graph below: There are different starting measurements from which one can solve a triangle, calculate the length of a side and height to it, and finally calculate a triangle's area. Geometry calculator for solving the Pythagorean Theorem of an right triangle given the length of a sides a and b. The Law of Cosines for angle C will be written as. The area of triangle was 30 sq.unit having length of base equal to 5 units. check out the law of sines ambiguous case Another way to calculate the exterior angle of a triangle is to subtract the angle of the vertex of interest from 180°. AAS - a side, 1 adjacent angle, and the opposite angle. ha=220, hb=165 hc=132... triangle calc by three heights. The sides of a right triangle are commonly referred to with the variables a, b, and c, where c is the hypotenuse and a and b are the lengths of the shorter sides. Want to understand the “Why” behind the “What”? You may find the manual calculation of circumcenter very difficult because it involves complicated equations and concepts. To find the missing sides or angles of the right triangle, all you need to do is enter the known variables into the trigonometry calculator. Applying in the above formula she will get. $$\dfrac{10}{\sin 56.09} = \dfrac{c}{\sin 93.91}$$, c = $$\dfrac{10 \times \sin 93.91}{\sin 56.09}$$. find the square value of side c. find the square value of side a. Subtract c^2 from a^2. What Are the General Formulae to Calculate "C" in a Triangle? Let's try to apply our learning on Heron's formula using the below given simulation. The Triangle Mastercard and the Triangle World Elite Mastercard do not have an annual fee. Claire drew a triangle with 2 sides and 3 angles as given below. It makes the process convenient by providing results on one click. Triangle Equations Formulas Calculator Mathematics - Geometry. SAS - 2 sides and the included angle given. Male or Female ? triangle calc, if know side, angle, and area of a triangle. error warnings. The variables a, b are the lengths of the shorter sides, also called legs or arms. Right Triangle Trig Calculator Fill in two values and press Calculate. Step #4: Tap the "Calculate Unknown" button. Input value you know and select what to compute. Find the root square value of the difference is the value of b. area S . a comprehensive calculator for triangles to solve angles and sides in an easy way Calculate missing parts of a triangle Select 3 of these elements and type in data. About this page: Triangle and area of a triangle calculator The calculator uses the Sine Law [ a ⁄ sin α = b ⁄ sin β = c ⁄ sin γ] to calculate the second angle of a triangle when two sides and an angle opposite one of them are given.Then, the calculator uses the projection rule to calculate the third side of a triangle: c = a cos β + b cos α. How will Jenny prove that when the Law of Cosines is applied to the right angled triangle, it'll result in formula of Pythagorean Theorem? 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# Independence of Random Variables II
Suppose $X$ and $Y$ are i.i.d. random variables. Also suppose they take the values from the set $\{1,2, \dots, n \}$. Then does this mean that $$P(X=1, Y= 1) = P(X=1) \cdot P(Y=1)$$ $$P(X=1, Y=2) = P(X=1) \cdot P(Y=2) \dots$$
So there are are $\binom{n}{2}$ cases for which $P(X \cap Y) =P(X) \cdot P(Y)$? If we didn't know that they were independent, we would have to check all these cases?
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@Henry: I know that they are i.i.d. I would just like to confirm that they are by going through the process. So there would be $\binom{n}{2}$ calculations? – alexm Mar 12 '12 at 14:34
For i.i.d. discrete random variables $X$ and $Y$, $(X,Y) \in \{1,2,\ldots, n\}^2$ has $n^2$ different points where $P\{X = i, Y = j\} = P\{X = i\}P\{Y = j\}$ holds, $i, j \in \{1,2,\ldots, n\}$, not $\binom{n}{2}$ points. To prove dependence, all you need to do is find one $(i,j)$ such that $P\{X = i, Y = j\} \neq P\{X = i\}P\{Y = j\}$. To prove that a given joint pmf corresponds to i.i.d random variables, you need to work harder. – Dilip Sarwate Mar 12 '12 at 14:43
Knowing $P(X=1, Y= 2) = P(X=1) \cdot P(Y=2)$ does not tell you $P(X=2, Y= 1) = P(X=2) \cdot P(Y=1)$ unless you have extra information.
So if you do not know that they are independent then you have to check almost $n^2$ pairs, though you can save a small number since probabilities add up to $1$.
To show they are identically distributed, you might also want to check $P(X=3)= P(Y=3)$ etc.
If we did know some information that guaranteed "symmetry", then we would have to check almost $\binom{n}{2}$ pairs? – alexm Mar 12 '12 at 14:39
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3.1: Formula Mass and the Mole Concept
Learning Objectives
• Calculate formula masses for covalent and ionic compounds
• Define the amount unit mole and the related quantity Avogadro’s number
• Explain the relation between mass, moles, and numbers of atoms or molecules, and perform calculations deriving these quantities from one another
We can argue that modern chemical science began when scientists started exploring the quantitative as well as the qualitative aspects of chemistry. For example, Dalton’s atomic theory was an attempt to explain the results of measurements that allowed him to calculate the relative masses of elements combined in various compounds. Understanding the relationship between the masses of atoms and the chemical formulas of compounds allows us to quantitatively describe the composition of substances.
Formula Mass
In an earlier chapter, we described the development of the atomic mass unit, the concept of average atomic masses, and the use of chemical formulas to represent the elemental makeup of substances. These ideas can be extended to calculate the formula mass of a substance by summing the average atomic masses of all the atoms represented in the substance’s formula.
Formula Mass for Covalent Substances
For covalent substances, the formula represents the numbers and types of atoms composing a single molecule of the substance; therefore, the formula mass may be correctly referred to as a molecular mass. Consider chloroform (CHCl3), a covalent compound once used as a surgical anesthetic and now primarily used in the production of tetrafluoroethylene, the building block for the “anti-stick” polymer, Teflon. The molecular formula of chloroform indicates that a single molecule contains one carbon atom, one hydrogen atom, and three chlorine atoms. The average molecular mass of a chloroform molecule is therefore equal to the sum of the average atomic masses of these atoms. Figure $$\PageIndex{1}$$ outlines the calculations used to derive the molecular mass of chloroform, which is 119.37 amu.
Likewise, the molecular mass of an aspirin molecule, C9H8O4, is the sum of the atomic masses of nine carbon atoms, eight hydrogen atoms, and four oxygen atoms, which amounts to 180.15 amu (Figure $$\PageIndex{2}$$).
Example $$\PageIndex{1}$$: Computing Molecular Mass for a Covalent Compound
Ibuprofen, C13H18O2, is a covalent compound and the active ingredient in several popular nonprescription pain medications, such as Advil and Motrin. What is the molecular mass (amu) for this compound?
Solution
Molecules of this compound are comprised of 13 carbon atoms, 18 hydrogen atoms, and 2 oxygen atoms. Following the approach described above, the average molecular mass for this compound is therefore:
Exercise $$\PageIndex{1}$$
Acetaminophen, C8H9NO2, is a covalent compound and the active ingredient in several popular nonprescription pain medications, such as Tylenol. What is the molecular mass (amu) for this compound?
151.16 amu
Formula Mass for Ionic Compounds
Ionic compounds are composed of discrete cations and anions combined in ratios to yield electrically neutral bulk matter. The formula mass for an ionic compound is calculated in the same way as the formula mass for covalent compounds: by summing the average atomic masses of all the atoms in the compound’s formula. Keep in mind, however, that the formula for an ionic compound does not represent the composition of a discrete molecule, so it may not correctly be referred to as the “molecular mass.”
As an example, consider sodium chloride, NaCl, the chemical name for common table salt. Sodium chloride is an ionic compound composed of sodium cations, Na+, and chloride anions, Cl, combined in a 1:1 ratio. The formula mass for this compound is computed as 58.44 amu (Figure $$\PageIndex{3}$$).
Note that the average masses of neutral sodium and chlorine atoms were used in this computation, rather than the masses for sodium cations and chlorine anions. This approach is perfectly acceptable when computing the formula mass of an ionic compound. Even though a sodium cation has a slightly smaller mass than a sodium atom (since it is missing an electron), this difference will be offset by the fact that a chloride anion is slightly more massive than a chloride atom (due to the extra electron). Moreover, the mass of an electron is negligibly small with respect to the mass of a typical atom. Even when calculating the mass of an isolated ion, the missing or additional electrons can generally be ignored, since their contribution to the overall mass is negligible, reflected only in the nonsignificant digits that will be lost when the computed mass is properly rounded. The few exceptions to this guideline are very light ions derived from elements with precisely known atomic masses.
Example $$\PageIndex{2}$$: Computing Formula Mass for an Ionic Compound
Aluminum sulfate, Al2(SO4)3, is an ionic compound that is used in the manufacture of paper and in various water purification processes. What is the formula mass (amu) of this compound?
Solution
The formula for this compound indicates it contains Al3+ and SO42− ions combined in a 2:3 ratio. For purposes of computing a formula mass, it is helpful to rewrite the formula in the simpler format, Al2S3O12. Following the approach outlined above, the formula mass for this compound is calculated as follows:
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Exercise $$\PageIndex{2}$$
Calcium phosphate, $$\ce{Ca3(PO4)2}$$, is an ionic compound and a common anti-caking agent added to food products. What is the formula mass (amu) of calcium phosphate?
310.18 amu
The Mole
The identity of a substance is defined not only by the types of atoms or ions it contains, but by the quantity of each type of atom or ion. For example, water, $$\ce{H2O}$$, and hydrogen peroxide, $$\ce{H2O2}$$, are alike in that their respective molecules are composed of hydrogen and oxygen atoms. However, because a hydrogen peroxide molecule contains two oxygen atoms, as opposed to the water molecule, which has only one, the two substances exhibit very different properties. Today, we possess sophisticated instruments that allow the direct measurement of these defining microscopic traits; however, the same traits were originally derived from the measurement of macroscopic properties (the masses and volumes of bulk quantities of matter) using relatively simple tools (balances and volumetric glassware). This experimental approach required the introduction of a new unit for amount of substances, the mole, which remains indispensable in modern chemical science.
The mole is an amount unit similar to familiar units like pair, dozen, gross, etc. It provides a specific measure of the number of atoms or molecules in a bulk sample of matter. A mole is defined as the amount of substance containing the same number of discrete entities (such as atoms, molecules, and ions) as the number of atoms in a sample of pure 12C weighing exactly 12 g. One Latin connotation for the word “mole” is “large mass” or “bulk,” which is consistent with its use as the name for this unit. The mole provides a link between an easily measured macroscopic property, bulk mass, and an extremely important fundamental property, number of atoms, molecules, and so forth.
The number of entities composing a mole has been experimentally determined to be $$6.02214179 \times 10^{23}$$, a fundamental constant named Avogadro’s number ($$N_A$$) or the Avogadro constant in honor of Italian scientist Amedeo Avogadro. This constant is properly reported with an explicit unit of “per mole,” a conveniently rounded version being $$6.022 \times 10^{23}/\ce{mol}$$.
Consistent with its definition as an amount unit, 1 mole of any element contains the same number of atoms as 1 mole of any other element. The masses of 1 mole of different elements, however, are different, since the masses of the individual atoms are drastically different. The molar mass of an element (or compound) is the mass in grams of 1 mole of that substance, a property expressed in units of grams per mole (g/mol) (Figure $$\PageIndex{4}$$).
Because the definitions of both the mole and the atomic mass unit are based on the same reference substance, 12C, the molar mass of any substance is numerically equivalent to its atomic or formula weight in amu. Per the amu definition, a single 12C atom weighs 12 amu (its atomic mass is 12 amu). According to the definition of the mole, 12 g of 12C contains 1 mole of 12C atoms (its molar mass is 12 g/mol). This relationship holds for all elements, since their atomic masses are measured relative to that of the amu-reference substance, 12C. Extending this principle, the molar mass of a compound in grams is likewise numerically equivalent to its formula mass in amu (Figure $$\PageIndex{5}$$).
Table $$\PageIndex{1}$$: Mass of one mole of elements
Element Average Atomic Mass (amu) Molar Mass (g/mol) Atoms/Mole
C 12.01 12.01 $$6.022 \times 10^{23}$$
H 1.008 1.008 $$6.022 \times 10^{23}$$
O 16.00 16.00 $$6.022 \times 10^{23}$$
Na 22.99 22.99 $$6.022 \times 10^{23}$$
Cl 33.45 35.45 $$6.022 \times 10^{23}$$
While atomic mass and molar mass are numerically equivalent, keep in mind that they are vastly different in terms of scale, as represented by the vast difference in the magnitudes of their respective units (amu versus g). To appreciate the enormity of the mole, consider a small drop of water after a rainfall. Although this represents just a tiny fraction of 1 mole of water (~18 g), it contains more water molecules than can be clearly imagined. If the molecules were distributed equally among the roughly seven billion people on earth, each person would receive more than 100 billion molecules.
The relationships between formula mass, the mole, and Avogadro’s number can be applied to compute various quantities that describe the composition of substances and compounds. For example, if we know the mass and chemical composition of a substance, we can determine the number of moles and calculate number of atoms or molecules in the sample. Likewise, if we know the number of moles of a substance, we can derive the number of atoms or molecules and calculate the substance’s mass.
Example $$\PageIndex{3}$$: Deriving Moles from Grams for an Element
According to nutritional guidelines from the US Department of Agriculture, the estimated average requirement for dietary potassium is 4.7 g. What is the estimated average requirement of potassium in moles?
Solution
The mass of K is provided, and the corresponding amount of K in moles is requested. Referring to the periodic table, the atomic mass of K is 39.10 amu, and so its molar mass is 39.10 g/mol. The given mass of K (4.7 g) is a bit more than one-tenth the molar mass (39.10 g), so a reasonable “ballpark” estimate of the number of moles would be slightly greater than 0.1 mol.
The molar amount of a substance may be calculated by dividing its mass (g) by its molar mass (g/mol):
The factor-label method supports this mathematical approach since the unit “g” cancels and the answer has units of “mol:”
$\mathrm{4.7\; \cancel{g} K \left ( \dfrac{mol\; K}{39.10\;\cancel{g}}\right)=0.12\;mol\; K} \nonumber$
The calculated magnitude (0.12 mol K) is consistent with our ballpark expectation, since it is a bit greater than 0.1 mol.
Exercise $$\PageIndex{3}$$: Beryllium
Beryllium is a light metal used to fabricate transparent X-ray windows for medical imaging instruments. How many moles of Be are in a thin-foil window weighing 3.24 g?
0.360 mol
Example $$\PageIndex{4}$$: Deriving Grams from Moles for an Element
A liter of air contains $$9.2 \times 10^{−4}$$ mol argon. What is the mass of Ar in a liter of air?
Solution
The molar amount of Ar is provided and must be used to derive the corresponding mass in grams. Since the amount of Ar is less than 1 mole, the mass will be less than the mass of 1 mole of Ar, approximately 40 g. The molar amount in question is approximately one-one thousandth (~10−3) of a mole, and so the corresponding mass should be roughly one-one thousandth of the molar mass (~0.04 g):
In this case, logic dictates (and the factor-label method supports) multiplying the provided amount (mol) by the molar mass (g/mol):
$\mathrm{9.2 \times10^{-4}\; \cancel{mol} \; Ar \left( \dfrac{39.95\;g}{\cancel{mol}\;Ar} \right)=0.037\;g\; Ar} \nonumber$
The result is in agreement with our expectations, around 0.04 g Ar.
Exercise $$\PageIndex{4}$$
What is the mass of 2.561 mol of gold?
504.4 g
Example $$\PageIndex{6}$$: Deriving Number of Atoms from Mass for an Element
Copper is commonly used to fabricate electrical wire (Figure $$\PageIndex{6}$$). How many copper atoms are in 5.00 g of copper wire?
Solution
The number of Cu atoms in the wire may be conveniently derived from its mass by a two-step computation: first calculating the molar amount of Cu, and then using Avogadro’s number (NA) to convert this molar amount to number of Cu atoms:
Considering that the provided sample mass (5.00 g) is a little less than one-tenth the mass of 1 mole of Cu (~64 g), a reasonable estimate for the number of atoms in the sample would be on the order of one-tenth NA, or approximately 1022 Cu atoms. Carrying out the two-step computation yields:
$\mathrm{5.00\:\cancel{g}\:Cu\left(\dfrac{\cancel{mol}\:Cu}{63.55\:\cancel{g}}\right)\left(\dfrac{6.022\times10^{23}\:atoms}{\cancel{mol}}\right)=4.74\times10^{22}\:atoms\: of\: copper}$
The factor-label method yields the desired cancellation of units, and the computed result is on the order of 1022 as expected.
Exercise $$\PageIndex{6}$$
A prospector panning for gold in a river collects 15.00 g of pure gold. How many Au atoms are in this quantity of gold?
$$4.586 \times 10^{22}\; Au$$ atoms
Example $$\PageIndex{7}$$: Deriving Moles from Grams for a Compound
Our bodies synthesize protein from amino acids. One of these amino acids is glycine, which has the molecular formula C2H5O2N. How many moles of glycine molecules are contained in 28.35 g of glycine?
Solution
We can derive the number of moles of a compound from its mass following the same procedure we used for an element in Example $$\PageIndex{6}$$:
The molar mass of glycine is required for this calculation, and it is computed in the same fashion as its molecular mass. One mole of glycine, C2H5O2N, contains 2 moles of carbon, 5 moles of hydrogen, 2 moles of oxygen, and 1 mole of nitrogen:
The provided mass of glycine (~28 g) is a bit more than one-third the molar mass (~75 g/mol), so we would expect the computed result to be a bit greater than one-third of a mole (~0.33 mol). Dividing the compound’s mass by its molar mass yields:
$\mathrm{28.35\:\cancel{g}\:glycine\left(\dfrac{mol\: glycine}{75.07\:\cancel{g}}\right)=0.378\:mol\: glycine} \nonumber$
This result is consistent with our rough estimate.
Exercise $$\PageIndex{7}$$
How many moles of sucrose, $$C_{12}H_{22}O_{11}$$, are in a 25-g sample of sucrose?
0.073 mol
Example $$\PageIndex{8}$$: Deriving Grams from Moles for a Compound
Vitamin C is a covalent compound with the molecular formula C6H8O6. The recommended daily dietary allowance of vitamin C for children aged 4–8 years is 1.42 × 10−4 mol. What is the mass of this allowance in grams?
Solution
As for elements, the mass of a compound can be derived from its molar amount as shown:
The molar mass for this compound is computed to be 176.124 g/mol. The given number of moles is a very small fraction of a mole (~10−4 or one-ten thousandth); therefore, we would expect the corresponding mass to be about one-ten thousandth of the molar mass (~0.02 g). Performing the calculation, we get:
$\mathrm{1.42\times10^{-4}\:\cancel{mol}\:vitamin\: C\left(\dfrac{176.124\:g}{\cancel{mol}\:vitamin\: C}\right)=0.0250\:g\: vitamin\: C} \nonumber$
This is consistent with the anticipated result.
Exercise $$\PageIndex{8}$$
What is the mass of 0.443 mol of hydrazine, $$N_2H_4$$?
14.2 g
Example $$\PageIndex{9}$$: Deriving the Number of Molecules from the Compound Mass
A packet of an artificial sweetener contains 40.0 mg of saccharin (C7H5NO3S), which has the structural formula:
Given that saccharin has a molar mass of 183.18 g/mol, how many saccharin molecules are in a 40.0-mg (0.0400-g) sample of saccharin? How many carbon atoms are in the same sample?
Solution
The number of molecules in a given mass of compound is computed by first deriving the number of moles, as demonstrated in Example $$\PageIndex{8}$$, and then multiplying by Avogadro’s number:
Using the provided mass and molar mass for saccharin yields:
$\mathrm{0.0400\:\cancel{g}\:\ce{C7H5NO3S}\left(\dfrac{\cancel{mol}\:\ce{C7H5NO3S}}{183.18\:\cancel{g}\:\ce{C7H5NO3S}}\right)\left(\dfrac{6.022\times10^{23}\:\ce{C7H5NO3S}\:molecules}{1\:\cancel{mol}\:\ce{C7H5NO3S}}\right)}\\ =\mathrm{1.31\times10^{20}\:\ce{C7H5NO3S}\:molecules}$
The compound’s formula shows that each molecule contains seven carbon atoms, and so the number of C atoms in the provided sample is:
$\mathrm{1.31\times10^{20}\:\ce{C7H5NO3S}\: molecules\left(\dfrac{7\:C\: atoms}{1\:\ce{C7H5NO3S}\: molecule}\right)=9.20\times10^{21}\:C\: atoms} \nonumber$
Exercise $$\PageIndex{9}$$
How many $$C_4H_{10}$$ molecules are contained in 9.213 g of this compound? How many hydrogen atoms?
• $$9.545 \times 10^{22}\; \text{molecules}\; C_4H_{10}$$
• $$9.545 \times 10^{23 }\;\text{atoms}\; H$$
Summary
The formula mass of a substance is the sum of the average atomic masses of each atom represented in the chemical formula and is expressed in atomic mass units. The formula mass of a covalent compound is also called the molecular mass. A convenient amount unit for expressing very large numbers of atoms or molecules is the mole. Experimental measurements have determined the number of entities composing 1 mole of substance to be 6.022 × 1023, a quantity called Avogadro’s number. The mass in grams of 1 mole of substance is its molar mass. Due to the use of the same reference substance in defining the atomic mass unit and the mole, the formula mass (amu) and molar mass (g/mol) for any substance are numerically equivalent (for example, one H2O molecule weighs approximately18 amu and 1 mole of H2O molecules weighs approximately 18 g).
Footnotes
1. 1 Omiatek, Donna M., Amanda J. Bressler, Ann-Sofie Cans, Anne M. Andrews, Michael L. Heien, and Andrew G. Ewing. “The Real Catecholamine Content of Secretory Vesicles in the CNS Revealed by Electrochemical Cytometry.” Scientific Report 3 (2013): 1447, accessed January 14, 2015, doi:10.1038/srep01447.
Glossary
experimentally determined value of the number of entities comprising 1 mole of substance, equal to 6.022 × 1023 mol−1
formula mass
sum of the average masses for all atoms represented in a chemical formula; for covalent compounds, this is also the molecular mass
mole
amount of substance containing the same number of atoms, molecules, ions, or other entities as the number of atoms in exactly 12 grams of 12C
molar mass
mass in grams of 1 mole of a substance
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1. ## 3D question
A pole, base at O, is on the west bank of a river. Two points, A and B, stand on the east bank of the river whose banks run parallel north to south. A to the south of O and B to the north. the distance AB is [tex]2a\sqrt{7}[/math and the angle AOB is $\displaystyle 150^o$. If the angle of elevation of the top of the pole, P, from A is $\displaystyle 45^o$ and $\displaystyle 30^o$ from B, find in terms of a the height of the pole and the width of the river.
I know OP=OA and OA=$\displaystyle 4a\sqrt{7}\sin B$
and $\displaystyle (A+B)=30^o$ but i don't know how to eliminate the sinB
thanks
2. Originally Posted by arze
I know OP=OA and OA=$\displaystyle 4a\sqrt{7}\sin B$
and $\displaystyle (A+B)=30^o$ but i don't know how to eliminate the sinB
thanks
OB = $\displaystyle 4a\sqrt{7}\sin A$ = sqrt{3}OP
OA=$\displaystyle 4a\sqrt{7}\sin B$ = OP
Sin(A)/sin(B) = sqrt{3}. But A = (30 - B)
So sin(30 - B)/sin(B) = sqrt{3}. expand sin(30-B)
(sin30*cos(B) - cos30*sinB)/sin(B) = sqrt{3}
1/2*cot(B) - sqrt{3}/2 = sqrt{3}
Solve the equation and find the angle B.
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# 10.12 kg to lbs - 10.12 kilograms into pounds
Do you want to learn how much is 10.12 kg equal to lbs and how to convert 10.12 kg to lbs? You are in the right place. In this article you will find everything about kilogram to pound conversion - theoretical and also practical. It is also needed/We also want to emphasize that all this article is dedicated to only one number of kilograms - exactly one kilogram. So if you want to learn more about 10.12 kg to pound conversion - keep reading.
Before we get to the more practical part - it means 10.12 kg how much lbs calculation - we will tell you a little bit of theoretical information about these two units - kilograms and pounds. So let’s start.
How to convert 10.12 kg to lbs? 10.12 kilograms it is equal 22.3107809144 pounds, so 10.12 kg is equal 22.3107809144 lbs.
## 10.12 kgs in pounds
We are going to start with the kilogram. The kilogram is a unit of mass. It is a basic unit in a metric system, in formal International System of Units (in abbreviated form SI).
Sometimes the kilogram could be written as kilogramme. The symbol of the kilogram is kg.
First definition of a kilogram was formulated in 1795. The kilogram was defined as the mass of one liter of water. First definition was simply but totally impractical to use.
Then, in 1889 the kilogram was described by the International Prototype of the Kilogram (in short form IPK). The International Prototype of the Kilogram was prepared of 90% platinum and 10 % iridium. The IPK was in use until 2019, when it was switched by another definition.
The new definition of the kilogram is based on physical constants, especially Planck constant. The official definition is: “The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015×10−34 when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where the metre and the second are defined in terms of c and ΔνCs.”
One kilogram is equal 0.001 tonne. It can be also divided into 100 decagrams and 1000 grams.
## 10.12 kilogram to pounds
You learned some information about kilogram, so now let's go to the pound. The pound is also a unit of mass. It is needed to point out that there are more than one kind of pound. What are we talking about? For instance, there are also pound-force. In this article we want to centre only on pound-mass.
The pound is in use in the Imperial and United States customary systems of measurements. Of course, this unit is in use also in other systems. The symbol of the pound is lb or “.
There is no descriptive definition of the international avoirdupois pound. It is defined as 0.45359237 kilograms. One avoirdupois pound is divided to 16 avoirdupois ounces or 7000 grains.
The avoirdupois pound was enforced in the Weights and Measures Act 1963. The definition of this unit was placed in first section of this act: “The yard or the metre shall be the unit of measurement of length and the pound or the kilogram shall be the unit of measurement of mass by reference to which any measurement involving a measurement of length or mass shall be made in the United Kingdom; and- (a) the yard shall be 0.9144 metre exactly; (b) the pound shall be 0.45359237 kilogram exactly.”
### How many lbs is 10.12 kg?
10.12 kilogram is equal to 22.3107809144 pounds. If You want convert kilograms to pounds, multiply the kilogram value by 2.2046226218.
### 10.12 kg in lbs
The most theoretical section is already behind us. In next part we are going to tell you how much is 10.12 kg to lbs. Now you learned that 10.12 kg = x lbs. So it is high time to get the answer. Have a look:
10.12 kilogram = 22.3107809144 pounds.
That is an accurate result of how much 10.12 kg to pound. It is possible to also round it off. After it your result is exactly: 10.12 kg = 22.264 lbs.
You learned 10.12 kg is how many lbs, so look how many kg 10.12 lbs: 10.12 pound = 0.45359237 kilograms.
Of course, in this case you can also round off this result. After it your result will be exactly: 10.12 lb = 0.45 kgs.
We are also going to show you 10.12 kg to how many pounds and 10.12 pound how many kg results in charts. Look:
We want to begin with a table for how much is 10.12 kg equal to pound.
### 10.12 Kilograms to Pounds conversion table
Kilograms (kg) Pounds (lb) Pounds (lbs) (rounded off to two decimal places)
10.12 22.3107809144 22.2640
Now look at a chart for how many kilograms 10.12 pounds.
Pounds Kilograms Kilograms (rounded off to two decimal places
10.12 0.45359237 0.45
Now you know how many 10.12 kg to lbs and how many kilograms 10.12 pound, so we can move on to the 10.12 kg to lbs formula.
### 10.12 kg to pounds
To convert 10.12 kg to us lbs you need a formula. We are going to show you two versions of a formula. Let’s begin with the first one:
Amount of kilograms * 2.20462262 = the 22.3107809144 result in pounds
The first formula will give you the most exact outcome. In some situations even the smallest difference could be considerable. So if you need a correct outcome - this formula will be the best for you/option to convert how many pounds are equivalent to 10.12 kilogram.
So let’s move on to the another version of a formula, which also enables calculations to know how much 10.12 kilogram in pounds.
The shorter formula is down below, have a look:
Number of kilograms * 2.2 = the outcome in pounds
As you see, the second version is simpler. It could be better solution if you need to make a conversion of 10.12 kilogram to pounds in easy way, for example, during shopping. You only need to remember that your outcome will be not so correct.
Now we want to learn you how to use these two formulas in practice. But before we will make a conversion of 10.12 kg to lbs we want to show you easier way to know 10.12 kg to how many lbs without any effort.
### 10.12 kg to lbs converter
Another way to know what is 10.12 kilogram equal to in pounds is to use 10.12 kg lbs calculator. What is a kg to lb converter?
Calculator is an application. Calculator is based on first formula which we gave you above. Due to 10.12 kg pound calculator you can quickly convert 10.12 kg to lbs. You only need to enter amount of kilograms which you want to convert and click ‘calculate’ button. The result will be shown in a flash.
So let’s try to calculate 10.12 kg into lbs using 10.12 kg vs pound calculator. We entered 10.12 as a number of kilograms. It is the result: 10.12 kilogram = 22.3107809144 pounds.
As you can see, our 10.12 kg vs lbs converter is so simply to use.
Now we can move on to our main issue - how to convert 10.12 kilograms to pounds on your own.
#### 10.12 kg to lbs conversion
We are going to begin 10.12 kilogram equals to how many pounds conversion with the first formula to get the most accurate outcome. A quick reminder of a formula:
Number of kilograms * 2.20462262 = 22.3107809144 the result in pounds
So what need you do to learn how many pounds equal to 10.12 kilogram? Just multiply number of kilograms, in this case 10.12, by 2.20462262. It is exactly 22.3107809144. So 10.12 kilogram is exactly 22.3107809144.
You can also round off this result, for example, to two decimal places. It is 2.20. So 10.12 kilogram = 22.2640 pounds.
It is high time for an example from everyday life. Let’s calculate 10.12 kg gold in pounds. So 10.12 kg equal to how many lbs? As in the previous example - multiply 10.12 by 2.20462262. It is 22.3107809144. So equivalent of 10.12 kilograms to pounds, when it comes to gold, is exactly 22.3107809144.
In this case it is also possible to round off the result. This is the outcome after rounding off, in this case to one decimal place - 10.12 kilogram 22.264 pounds.
Now we can move on to examples converted with short formula.
#### How many 10.12 kg to lbs
Before we show you an example - a quick reminder of shorter formula:
Number of kilograms * 2.2 = 22.264 the outcome in pounds
So 10.12 kg equal to how much lbs? As in the previous example you need to multiply number of kilogram, in this case 10.12, by 2.2. Let’s see: 10.12 * 2.2 = 22.264. So 10.12 kilogram is equal 2.2 pounds.
Let’s make another calculation using this formula. Now convert something from everyday life, for instance, 10.12 kg to lbs weight of strawberries.
So convert - 10.12 kilogram of strawberries * 2.2 = 22.264 pounds of strawberries. So 10.12 kg to pound mass is exactly 22.264.
If you know how much is 10.12 kilogram weight in pounds and can calculate it with use of two different formulas, we can move on. Now we are going to show you all results in charts.
#### Convert 10.12 kilogram to pounds
We realize that results presented in charts are so much clearer for most of you. We understand it, so we gathered all these outcomes in tables for your convenience. Due to this you can easily make a comparison 10.12 kg equivalent to lbs outcomes.
Let’s begin with a 10.12 kg equals lbs chart for the first formula:
Kilograms Pounds Pounds (after rounding off to two decimal places)
10.12 22.3107809144 22.2640
And now have a look at 10.12 kg equal pound chart for the second formula:
Kilograms Pounds
10.12 22.264
As you see, after rounding off, when it comes to how much 10.12 kilogram equals pounds, the outcomes are the same. The bigger number the more considerable difference. Please note it when you need to do bigger amount than 10.12 kilograms pounds conversion.
#### How many kilograms 10.12 pound
Now you know how to calculate 10.12 kilograms how much pounds but we will show you something more. Do you want to know what it is? What about 10.12 kilogram to pounds and ounces conversion?
We are going to show you how you can calculate it step by step. Let’s start. How much is 10.12 kg in lbs and oz?
First things first - you need to multiply number of kilograms, this time 10.12, by 2.20462262. So 10.12 * 2.20462262 = 22.3107809144. One kilogram is equal 2.20462262 pounds.
The integer part is number of pounds. So in this example there are 2 pounds.
To calculate how much 10.12 kilogram is equal to pounds and ounces you have to multiply fraction part by 16. So multiply 20462262 by 16. It is exactly 327396192 ounces.
So your result is 2 pounds and 327396192 ounces. It is also possible to round off ounces, for instance, to two places. Then your result is equal 2 pounds and 33 ounces.
As you can see, calculation 10.12 kilogram in pounds and ounces quite simply.
The last calculation which we will show you is conversion of 10.12 foot pounds to kilograms meters. Both foot pounds and kilograms meters are units of work.
To calculate foot pounds to kilogram meters it is needed another formula. Before we give you this formula, have a look:
• 10.12 kilograms meters = 7.23301385 foot pounds,
• 10.12 foot pounds = 0.13825495 kilograms meters.
Now see a formula:
Amount.RandomElement()) of foot pounds * 0.13825495 = the result in kilograms meters
So to convert 10.12 foot pounds to kilograms meters you need to multiply 10.12 by 0.13825495. It is 0.13825495. So 10.12 foot pounds is exactly 0.13825495 kilogram meters.
You can also round off this result, for example, to two decimal places. Then 10.12 foot pounds will be exactly 0.14 kilogram meters.
We hope that this calculation was as easy as 10.12 kilogram into pounds calculations.
This article was a big compendium about kilogram, pound and 10.12 kg to lbs in calculation. Due to this calculation you learned 10.12 kilogram is equivalent to how many pounds.
We showed you not only how to do a conversion 10.12 kilogram to metric pounds but also two another calculations - to know how many 10.12 kg in pounds and ounces and how many 10.12 foot pounds to kilograms meters.
We showed you also another solution to do 10.12 kilogram how many pounds conversions, it is with use of 10.12 kg en pound calculator. It is the best choice for those of you who do not like calculating on your own at all or need to make @baseAmountStr kg how lbs conversions in quicker way.
We hope that now all of you are able to do 10.12 kilogram equal to how many pounds conversion - on your own or with use of our 10.12 kgs to pounds calculator.
It is time to make your move! Calculate 10.12 kilogram mass to pounds in the way you like.
Do you want to make other than 10.12 kilogram as pounds calculation? For example, for 15 kilograms? Check our other articles! We guarantee that conversions for other numbers of kilograms are so easy as for 10.12 kilogram equal many pounds.
### How much is 10.12 kg in pounds
At the end, we are going to summarize the topic of this article, that is how much is 10.12 kg in pounds , we prepared one more section. Here we have for you all you need to remember about how much is 10.12 kg equal to lbs and how to convert 10.12 kg to lbs . Let’s see.
How does the kilogram to pound conversion look? To make the kg to lb conversion it is needed to multiply 2 numbers. How does 10.12 kg to pound conversion formula look? . Check it down below:
The number of kilograms * 2.20462262 = the result in pounds
Now you can see the result of the conversion of 10.12 kilogram to pounds. The correct answer is 22.3107809144 pounds.
It is also possible to calculate how much 10.12 kilogram is equal to pounds with second, easier version of the equation. Check it down below.
The number of kilograms * 2.2 = the result in pounds
So now, 10.12 kg equal to how much lbs ? The result is 22.3107809144 pounds.
How to convert 10.12 kg to lbs in just a moment? It is possible to use the 10.12 kg to lbs converter , which will make whole mathematical operation for you and you will get an accurate answer .
#### Kilograms [kg]
The kilogram, or kilogramme, is the base unit of weight in the Metric system. It is the approximate weight of a cube of water 10 centimeters on a side.
#### Pounds [lbs]
A pound is a unit of weight commonly used in the United States and the British commonwealths. A pound is defined as exactly 0.45359237 kilograms.
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# Question: What Is Snell’S Law Of Refraction Class 10?
## What is Snell’s law class 10th?
Snell’s law is defined as “The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant, for the light of a given colour and for the given pair of media”..
## What is the formula of refraction?
When scientists talk about refraction, they use a formula. “n = c / v” “c” is the speed of light in a vacuum, “v” is the speed of light in that substance and “n” is the index of refraction.
## What is sin i and sin r?
1. At the point of incidence, the incident ray, refracted ray and normal all lie in the same plane. … When light is travelling from air to a denser medium, the angle of incidence and angle of refraction are related by the ratio sin i / sin r = n whereby n is the refractive index of the denser medium.
## What is called refractive index?
Refractive Index (Index of Refraction) is a value calculated from the ratio of the speed of light in a vacuum to that in a second medium of greater density. The refractive index variable is most commonly symbolized by the letter n or n’ in descriptive text and mathematical equations.
## What is Snell’s law for?
Snell’s Law is a formula used to discribe the relationship between the angles of incidence and refraction,when referring to light or other waves passing through a boundary between to different isotropic media,such as water,glass and air.
## What are the 3 laws of refraction?
Laws of refraction state that: The incident ray, reflected ray and the normal, to the interface of any two given mediums; all lie in the same plane. The ratio of the sine of the angle of incidence and sine of the angle of refraction is constant.
## How do you test Snell’s law?
Verifying Snell’s Law Turn on the ray box and aim the light ray towards the glass block so that it makes an angle with the nearest surface of the block as shown in the picture. For each piece of paper, change the angle of the incoming ray.
## What are two types of refractive index?
Relative refractive index– It is the ratio of speed of light in one medium to the speed of light in another medium • Absolute refractive index– It is the ratio of light in vacuum to the speed of light in another medium.
## How do you calculate sin R?
Calculate the refractive index.Work out the sine of angle i. sin 55 = 0.819.Work out the sine of angle r. sin 33 = 0.545.Divide sin i by sin r. refractive index = sin i ÷ sin r. refractive index = 0.819 ÷ 0.545 = 1.50. A practical demonstration of how Pyrex seems to disappear in vegetable oil. Page 4 of 4. Move on to Test.
## What refraction means?
Medical definitions for refraction The turning or bending of any wave, such as a light or sound wave, when it passes from one medium into another of different density. The ability of the eye to bend light so that an image is focused on the retina.
## What is refractive index Class 10th?
Refractive index is a measure of how much speed of light changes when it enter the medium from air. Absolute refractive index is the ratio of speed of light in vacuum or air to speed of light in the medium. c. n=—— ν
## What is mean by Snell’s law?
the law that, for a ray incident on the interface of two media, the sine of the angle of incidence times the index of refraction of the first medium is equal to the sine of the angle of refraction times the index of refraction of the second medium.
## What is unit of refractive index?
The refractive index of a medium is the ratio of the speed of light in vacuum to the speed of light in the medium. It has no units, therefore. This is true also in the context of surface plasmon resonance. RIU is sometimes used to distinguish a number as referring to a refractive index.
## How do you calculate sin?
Sin, Cos and TanThe sine of the angle = the length of the opposite side. the length of the hypotenuse.The cosine of the angle = the length of the adjacent side. the length of the hypotenuse.The tangent of the angle = the length of the opposite side. the length of the adjacent side.
## Why is sin a sin R constant?
For any two given pair of mediums, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant.
## What is angle of refraction?
The angle that the incident ray makes with the normal line is referred to as the angle of incidence. Similarly, the angle that the refracted ray makes with the normal line is referred to as the angle of refraction. The angle of incidence and angle of refraction are denoted by the following symbols: = angle of incidence.
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What is horsepower and how is it utilized with servo motors
June 23, 2017, by Hurley Gill
Block and Tackle Series Volume 1 - Horsepower
Welcome to our Block and Tackle series on Blog in Motion! We went back to basics to help answer some questions covering a variety of topics. We pose the question – and then offer an answer. We welcome your feedback on other questions or topics you might have – just add a comment to this blog post and we’ll add it to our list. Here we go!
Question: What is horsepower (hp) and how is it utilized with servo motors?
Horsepower (hp) is a measure of power, which can be further described as the rate at which work is performed. There are slightly different definitions for its conversion to the unit watts depending on the mechanism being described: mechanical, electric, boiler, metric, etc.. Our focus here will be on servo motor systems.
The SI unit Watt for the measure of power has been adopted by most countries of the world. However, the unit hp often survives because it is typically utilized to specifically define an electric motor’s output capability, reducing potential confusion between watts input, watts output [hp], and watts demanded by a load, while defining a system.
Just a side note – the watt (W) is named after a Scottish engineer, James Watt (1736-1819). (Also, remember to use the capital letter for units that are named after a person; like amp (A), volts (V), etc.
For electric motors: 1 hp = 746 Watts.
The basic equation for calculating horsepower is:
The constant: 5252, is the rounded off calculation: 33,000 ft-lb/minute / 2π rad/revolution.
As the story goes, the standardized value of 33,000 ft-lb/minute was established in approximately 1783 by James Watt and Matthew Boulton based on the estimated output of a brewery horse
Since there are 1.35582 Nm/ ft-lb, the metric equivalent equation would be:
Horsepower (hp) defined by servo motor catalog data sheets is mostly used as a relative number for comparison purposes of continuous and/or peak capability when choosing a servo motor, since most applications require a knowledge of the required working torques and speeds.
Let us know if you have any questions on this or any other topic in motion!
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# Standard Deviation Calculator Online (Sample and Population)
The study of statistics is fascinating, and we want to help you discover this feeling. One of the especially useful topics is the standard deviation, as it gives you great information to observe and compare data. In this article, you will find our online standard deviation calculator, where you will learn how to obtain it step by step.
## What is Standard Deviation?
Standard deviation is a statistical measure of the dispersion or variation of a set of data. This dispersion tells us how the data are distributed about the mean.
To calculate the standard deviation, we will use the following formulas, depending on whether the data to evaluate corresponds to the population or if it only represents a sample:
#### (b) Sample standard deviation:
Where:
• σ: Population standard deviation.
• s: Sample standard deviation.
• xÌ„: Mean.
• N: Number of evaluated values.
• xi: Each of the values.
## How to use the online standard deviation calculator?
To calculate the standard deviation using our application, we will follow the following steps:
• Choose the decimal number notation and the data separator by selecting the corresponding options.
• Enter the set of values to evaluate.
• Click on Solve.
• Next, you will be able to visualize the detail of the calculations performed.
To understand it better, we will see its use with an example:
### Example of standard deviation calculation
Find the standard deviation of the following numbers:
12, 15, 17, 20, 30, 31, 43, 44, 54
### Solution
We enter the values in the tool:
When clicking on “Solve” we will obtain the following:
Population Standard Deviation: 13.9771
Sample Standard Deviation: 14.8249
Population Variance: 195.358
Sample Variance: 219.7778
Mean: 29.5556
Count of Data: 9
According to the data of the problem we have:
• Σxᵢ = 266
• NÂ =Â 9
• xÌ„Â = 266/9 = 29.5556
At the end, you will find the option to copy the problem link so that you can see the results without having to re-enter the data. It is beneficial to share the solutions with a colleague.
Finally, if you want to find other measures of dispersion or central tendency, you can visit the following options:
## Final Reflection
As you can see, knowing about standard deviation is extremely important for statistical analysis. Our calculator is an excellent tool that will help you deepen everything related to this statistical concept.
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Best answer: Is the gravitational attraction force between two objects on earth small or large?
Contents
But here’s the thing to remember: Gravitational force depends on the mass of the objects and the square of the distance between them (g = Gm1m2/r^2). So the gravitational force between earthly objects is low since the objects are miniscule compared to celestial objects.
Why are gravitational forces between masses on earth so small?
Since gravitational force is inversely proportional to the square of the separation distance between the two interacting objects, more separation distance will result in weaker gravitational forces. So as two objects are separated from each other, the force of gravitational attraction between them also decreases.
Is the gravitational force of attraction between the two objects the greatest?
The closer two objects are, the stronger the gravitational pull between them. So, putting these rules together, the more massive and the closer two objects are, the greater the gravitational attraction between them.
THIS IS UNIQUE: Best answer: What countries were included in the Grand Tour?
What is the gravitational force between two objects attractive at?
Gravitational force -an attractive force that exists between all objects with mass; an object with mass attracts another object with mass; the magnitude of the force is directly proportional to the masses of the two objects and inversely proportional to the square of the distance between the two objects.
How big is the gravitational force between you and the earth?
Earth’s Gravity
This results in Earth having a gravitational strength of 9.8 m/s² close to the surface (also known as 1 g), which naturally decreases the farther away one is from the surface. In addition, the force of gravity on Earth actually changes depending on where you’re standing on it.
What happens to the force of gravitational attraction between two small objects?
Since gravitational force is inversely proportional to the square of the separation distance between the two interacting objects, more separation distance will result in weaker gravitational forces. So as two objects are separated from each other, the force of gravitational attraction between them also decreases.
What is gravitational attraction?
Definitions of gravitational attraction. (physics) the force of attraction between all masses in the universe; especially the attraction of the earth’s mass for bodies near its surface. synonyms: gravitation, gravitational force, gravity.
Is the gravitational force between two objects the same?
The Law of Universal Gravitation
The force of gravitational attraction between any two massive bodies is proportional to their masses and inversely proportional to the square of the distance between their centers.
Do two objects exert a gravitational force on each other?
What Newton said was this: whenever there are two objects that have mass, they will exert a gravitational force on each other that is proportional to the product of the masses, and inversely proportional to the square of the distance between them.
What determines the gravitational force between objects?
Two factors determine the magnitude of the gravitational force between two objects: (1) their masses and (2) the separation distance between them. The size of the force is proportional to the product of the masses of the two objects.
What is the gravitational force between two objects attractive at small distances only?
Newton’s Law of Gravitation tells that gravitational attractive force between two objects is directly proportional to the product of the masses and inversely proportional to the square of the separation distance.
Is gravitational force attractive in nature?
Complete answer: We know that the gravitational force is defined as the multiplication of the gravitational constant along with the masses of the objects and then divided by the square of the distance between the objects. … Hence, the nature of the gravitational force will always be attractive and not repulsive.
What is the gravitational force between two small bodies?
This force of attraction is directly proportional to the product of the two masses involved and inversely proportional to the square of the distances between them. This law is known as the law of universal gravitation and was formulated by Sir Isaac Newton in 1687.
THIS IS UNIQUE: Frequent question: Can I stay in a country after my visa expires?
Does the gravitational force of attraction of earth become zero?
Ans; No. The acceleration due to gravity of the earth, decreases by going away from the surface of the earth. But it never becomes zero, although it gets weaker and weaker. In fact the acceleration due to gravity or gravitational force becomes zero at infinity.
Why are small objects not attracted to each other?
Newton’s universal law of gravitation states that all objects in the Universe attract all other objects. … The gravitational pull between small objects, such as molecules and books, is generally negligible; the gravitational pull exerted by larger objects, such as stars and planets, organizes the Universe.
How do you find the force of gravitational attraction?
The mathematical formula for gravitational force is F=GMmr2 F = G Mm r 2 where G is the gravitational constant.
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# Placement Questions & Answers :: TCS
## Total View: 63.85K
#### Co. Cloud
Not Attempted
471. How many 2's are there between the terms 112 to 375?
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Answer: Option C
Explanation:
Let us calculate total 2's in the units place. (122, 132, 142 ... 192), (201, 212, 222, ... 292), (302, 312, ... 372) = 8 + 10 + 8 = 26
Total 2's in tenth's place, (120, 121, 122, ..., 129) + (220, 221, ..., 229) + (320, 321, ..., 329) = 30
Total 2's in hundred's place = (200, 201, ... 299) = 100.
Total 2's between 112 and 375 = 26 + 30 + 100 = 156
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Tags: TCS
Not Attempted
472. Consider the sequence of numbers 0, 2, 2, 4,... Where for n > 2 the nth term of the sequence is the unit digit of the sum of the previous two terms. Let sn denote the sum of the first n terms of this sequence. What is the smallest value of n for which sn >2771?
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Answer: Option B
Explanation:
[0, 2, 2, 4, 6, 0, 6, 6, 2, 8, 0, 8, 8, 6, 4, 0, 4, 4, 8, 2], 0, 2, 2...this series repeats after every 20 terms.
Sum of these 20 terms = 80
So 2771 =34*80 + 51
Sum of 13 terms = 52
So we have to use 34 times 20 terms = 34*20 = 680
680+13 = 693
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Tags: TCS
Not Attempted
473. Three generous friends, each with some money, redistribute the money as follows: Sandra gives enough money to David and Mary to double the amount of money each has. David then gives enough to Sandra and Mary to double their amounts. Finally, Mary gives enough to Sandra and David to double their amounts. If Mary had 11 rupees at the beginning and 17 rupees at the end, what is the total amount that all three friends have?
View Answer | Submit Your Solution | Important Formulas | Topic: | |
Answer: Option D
Explanation:
Let Sandra, David and Mary each has s, d and 11 respectively.
After the first distribution,
David has d + d = 2d, Mary has 11 + 11 = 22 and Sandra has s - d - 11.
After the second distribution,
Sandra has 2*(s - d - 11) , mary has 2*22 = 44 and david has 2d - (s - d - 11) - 22=3d - s -11.
After the third distribution,
Sandra has 2*2(s - d - 11), david has 2*(3d - s - 11) and mary has 44 - 2(s - d - 11) - (3d - s - 11) = 77 - s - d
It is given that finally Mary has Rs.17. So, 77 - s - d=17
=> s + d = 60
=> s + d + 11 = 60 + 11 = 71.
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Not Attempted
474. Babla alone can do a piece of work in 10 days. Ashu alone can do it in 15 days. The total wages for the work is Rs.5000. How much should be Babla be paid if they work together for an entire duration of work.
View Answer | Submit Your Solution | Important Formulas | Topic: | |
Answer: Option D
Explanation:
Bablu work = 1/10
Ashu work = 1/15
Total work = 1/10+1/15 = 3+2/30 = 5/30
Total work = 6 days
Bablu paid = 6/10*5000 = 3000
Submit Your Solution
Tags: TCS
Not Attempted
475. In a group of five families, every family is expected to have a certain number of children, such that the number of children forms an arithmetic progression with a common difference of one, starting with two children in the first family. Despite the objection of their parents, every child in a family has as many pets to look after as the number of offsprings in the family. What is the total number of pets in the entire group of five families.
View Answer | Submit Your Solution | Important Formulas | Topic: | |
Answer: Option D
Explanation:
As the number of children are in arithmetic progression starting with 2, the five families have 2, 3, 4, 5, 6 kids respectively.
As each children has kept the pets equal to the number of kids in the family, Each family has n^2 pets.
So total = 2^2+3^2+4^2 +5^2 +6^2 = 90
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ShortCut
Not Attempted
476. Radius of the bigger circle is 1. Which area will be greater?
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Answer: Option B
Explanation:
Here is no explanation for this answer
ShortCut By :: Vikas Kumar
From the given options we can see that area marked with 4 is the answer.
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Not Attempted
477. An old man and a young man are working together in an office and staying together in a near by apartment. The old man takes 30 minutes and the young 20 minutes to walk from appartment to office. If one day the old man started at 10.00 AM and the young man at 10:05AM from the apartment to office, when will they meet?
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Answer: Option A
Explanation:
Ratio of old man speed to young man speed = 2:3
The distance covered by old man in 5 min = 10
The 10 unit is covered with relative speed=10/(3-2)=10 min
so, they will meet at 10:15 am.
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Tags: TCS
Not Attempted
478. In a potato race, 20 potatoes are placed in a line of intervals of 4 meters with the first potato 24 meters from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
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Answer: Option C
Explanation:
Given, total number of potatos = 20.
First potato 24 metres from the starting point. There are 4 meters in the intervals. A contestant is required to bring the potatoes back to the starting place one at a time. So for the first potato he has to travel 48 meters, for second 56 meters ...
48,56,64............20 terms.
a = 48, d= 8, n = 20.
Important Point:
Sum of n terms in A.P = Sn=n2[2a+(n?1)d]
S20 = 202[2*48+(20?1)8]
S20 = 202[96+152]
S20 = 10*248 = 2480
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Tags: TCS
Not Attempted
479. Let a, b, c, d and e be distinct integers in ascending order such that(76-a)(76-b)(76-c)(76-d)(76-e) = 1127. What is a + b + c + d
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Answer: Option B
Explanation:
Product of 5 terms equal to 1127. As all the five terms are integers, given product should be a product of 5 numbers. Now factorize 1127.
1127 = 72 * 23 = 7 * 7 * 23
But given that all the a, b, c, d, e are distinct. And we are getting only 3 terms with 7 repeats.
Now the logic is, integers means positive and negative, 7 and - 7 possible and 1, - 1 also possible .
As a,b, c, d, e are in ascending order, the factors should be in decreasing order. So (23, 7, 1, -1, -7)
Now a = 53; b = 69; c = 75; d = 77
a + b + c + d = 274.
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Tags: TCS
Not Attempted
480. In this question, A^B refers to A raised to the power B.Ten tickets numbered 1, 2, 3, ..., 10. Six tickets are selected at random one of a time with replacement. The probability of the largest number appearing on the selected ticket is 7 is
View Answer | Submit Your Solution | Important Formulas | Topic: | |
Answer: Option B
Explanation:
Let's first find out probability of that maximum number being any number between 1 to 7.
P(1 to 7) = (7/10) * (7/10) * (7/10) ... 6 times
Now find out probability of that maximum number being any number between 1 to 6.
P(1 to 6) = (6/10) * (6/10) * (6/10) ... 6 times
Now, probability that maximum number is exactly 7
= P(1 to 7) - P (1 to 6)
= (7^6 - 6^6) / 10^6
Submit Your Solution
Tags: TCS
Here is the list of questions asked in TCS Aptitude Test Question with Answers TCS Mock Test page 48. Practice TCS Written Test Papers with Solutions and take Q4Interview TCS Online Test Questions to crack TCS written round test. Overall the level of the TCS Online Assessment Test is moderate. Only those candidates who clear the written exam will qualify for the next round, so practic all the questions here and take all the free tests before going for final selection process of TCS
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# Introduction to Computer Architecture
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1 Boolean Operators The Boolean operators AND and OR are binary infix operators (that is, they take two arguments, and the operator appears between them.) A AND B D OR E We will form Boolean Functions of these by setting them equal to a result variable, f. f AND = A AND B f OR = D OR E They are formally defined by specifying the Truth Table associated with them. A truth table is a table in which all possible combinations of values for the arguments (sometimes called inputs, for reasons which will become clear later) are listed, and the value of the function (result, or output) for each combination is given. Since these are binary operators, there are two arguments, each of which can take on only the values of T or F, so that there are only a total of four possible combinations of values. Thus there are only four rows in the truth table, which is Arguments Result A B f AND F F F F T F T F F T T T This truth table defines the AND function by specifying that the result of the function is TRUE only if both of the arguments are TRUE. This should be intuitively make sense. In words, if A is TRUE AND B is TRUE, then f AND is TRUE, otherwise f AND is FALSE. Alternatively, it is sometimes useful to interpret the table as saying if either A is FALSE or B is FALSE, then f AND is FALSE, otherwise it is TRUE. In an identical fashion, the OR function, f OR, is defined by the truth table A B f OR 1
2 F F F F T T T F T T T T This truth table defines the OR function by specifying that the result of the function is TRUE whenever either of the arguments is TRUE. This should be intuitively make sense. In words, if A is TRUE OR B is TRUE, then f OR is TRUE, otherwise f OR is FALSE. Alternatively, it is sometimes useful to interpret the table as saying if both A is FALSE AND B is FALSE, then f OR is FALSE, otherwise it is TRUE. Notice that the OR function includes the case, and is TRUE, when both rguments are TRUE; for this reason it is sometimes referred to as the inclusive OR. We will discuss the exclusive OR shortly. The Boolean operator NOT is a monadic prefix operator, meaning it takes just one argument, and the operator precedes the argument. NOT A The corresponding function is f NOT = NOT A and the truth table is given as A F T f NOT T F That is, f NOT is the complement of A. We will now introduce some symbolism to make writing Boolean expressions more compact and easier to read (trust me). The AND and OR functions are represented by many different symbols in the literature, usually depending on the context of the discussion. For instance the following symbols are often used for the AND function:,, and. The symbols,, and + are frequently used for OR, and ~,, or an overbar are used for NOT. In what follows we will use the following: 2
3 A AND B will be given by A B or AB A OR B will be given by A+B NOT A will be given by A In addition, FALSE, or F, will be given by the binary digit 0 TRUE, or T will be given by the binary digit 1 Rewriting the truth tables above in this symbolism gives A B AB A B A+B A A Boolean Expressions A Boolean expression is any combination of Boolean variables, constants and operators. E.g. AB + C(A+BC) + BC [B1] Such an expression has a value (0 or 1) only when all of the variables are assigned values of either 0 or 1. Of course, the order in which the operations are done must be correct. This is determined, just as with arithmetic expressions, using a predefined order of precedence, modified by parentheses if necessary. Boolean expressions are evaluated left to right, with expressions within parentheses being evaluated first. Among the AND, OR and NOT operators, the order of precedence is NOT, AND, OR. Thus, in the above expression, A+BC is evaluated first (within parentheses), than the complement of that part of the expression is evaluated. Then the ANDs are evaluated, and finally, the outermost ORs are evaluated. Suppose, in this example, values are assigned to the variables thusly: A=1, B=0, and C=1. Then the value of expression [B1] will be 1. We will show how this is arrived at later. Without assigning values to the variables, there are still many important jobs that may need to be done with such expressions, including, as stated above, proving different Boolean Expressions equivalent, or simplifying them. 3
4 For example a) Simplify AB+AB C = AB+AC We will present two methods for simplification later. b) Prove ABC+ ABC +AB C is equivalent to AB + AC It is clear that a Boolean expression is not unique - there are many Boolean expressions which will evaluate to the same result for the same assignment of variable values. Such expressions are called equivalent. Truth Tables Just as with the fundamental functions AND, OR and NOT, any Boolean Expression can be fully described by its truth table. The expression [B1] has the following truth table (let s call the expression f B1 ) A B C f B Notice that if there are n variables in the expression, then its truth table must have 2 n rows, since this is the total number of ways 1's and 0's can be assigned to the variables. Two Boolean Expressions are equivalent if they have the exact same truth table - that is, the 1's in the f column of the truth table are in the exact same rows. Determining the Boolean Expression given the Truth table Consider the following truth table: A B f
5 The algebraic Boolean expression is determined as follows: 1. Consider only the rows where f = 1 2. For each such row, write a term containing all the variables; if the value of a variable in the row is 1, write the variable; if the value is 0, write the variable complemented. 3. The final expression is generated by taking the OR of all the terms generated in step 2. In the current example, only the first and fourth rows have 1's. From the first row we create the term A B (both variables have the value 0 in this row, so we use the complement of both variables.) Similarly, from the fourth row we get the term AB. The final expression, ORing these two terms together is Consider another example: f = A B + AB A B C f The Boolean expression corresponding to this table is f = A BC +A BC+AB C +AB C [B2] Any Boolean expression which is expressed as the OR of ANDs, as is [B2], is said to be in Sum of Products form. If, in addition, every one of the ANDs contains every variable, the form is a canonical one called the Disjunctive Normal Form (DNF). The method of generating Boolean expressions from truth tables just shown always produces the expression s DNF. Each term in [B2] is called a minterm. An alternative but equivalent expression from the same truth table can be expressed in a Product of Sums form f = (A+B+C)(A+B+C )(A +B +C)(A +B +C ) [B3] 5
6 Again, if every sum term in the expression contains all variables, then we have another canonical form called the Conjunctive Normal Form (CNF). Each term in [B3] is called a maxterm. As stated earlier, Boolean expressions are not unique for any given function. Note, for example, that the function [B2] could also be given (in sum of products form, but not DNF as) f = AB + A B (and yes, the variable C is not needed; that is, the value of C has no effect upon the value of f. Such terms are referred to as Don t Cares ) Practice problems - Deriving Boolean Expressions From Truth Tables: What is the Boolean expression for the following truth tables? 1. A B f b. A B C f Some Interesting Facts 1. There are 2 n minterms for a function of n variables. For instance, there are 2 2 =4 minterms of two variables: AB, AB, A B, A B and 2 3 = 8 minterms of 3 variables: ABC, ABC, AB C, AB C, A BC, A BC, A B C, A B C 6
7 2. All Boolean expressions of n variables can be found by taking all possible subsets of the minterms that exist for n variables. Thus, there are 2^2 n functions of n variables. For two variables, there are 2^2 2 = 16 functions, all of which are shown (in DNF form) here: f 0 =0 (no minterms), f 1 = AB f 2 = AB f 3 = A B f 4 = A B f 5 = AB + AB f 6 = AB + A B f 7 = AB + A B f 8 = AB + A B f 9 = AB + A B f 10 = A B + A B f 11 = AB + AB + A B f 12 = AB + AB + A B f 13 = AB + A B + A B f 14 = AB + A B + A B f 15 = AB + AB + A B + A B = 1 Note that f = 0 when no minterms are present and f = 1 when all minterms are present. 3. If f is represented in DNF form (such as all the examples in 2 above) then f is represented by all those terms not present in f. Examples: If f 5 = AB + AB then f 5 = A B + A B = f 10 If f = ABC + ABC + A B C then f = AB C + A BC + AB C + A BC + A B C In terms of a truth table, if f is the sum of all the minterms with a 1 in the f column, then f is the sum of all the minterms with a 0 in the f column. Practice Problems - Facts 1. How many minterms exist for a Boolean space of 5 variables? 2. How many possible Boolean functions of 4 variables are there? 3. List all minterms for a Boolean space of 1 variable. 4. List all functions for a boolean space of 1 variable 5. If f = A B, what is the DNF for f? Algebraic Rules of 7
8 Determining the equivalence of two Boolean expressions can be done algebraically as well as by using truth tables, assuming that we know the rules, postulates and theorems, of. In addition we can use such rules to simplify Boolean expressions, and to take arbitrary Boolean expressions and put them in canonical form (for comparison, or truth table generation.) Let s start with some axioms, or postulates (fundamental properties) of Boolean Algebra. Postulates 1. Both AND and OR have identity elements A + 0 = A A 1 = A 2. Commutativity A+B = B+A AB = BA 3. Associativity A+(B+C) = (A+B)+C A(BC) = (AB)C 4. Distributivity A+BC = (A+B)(A+C) A(B+C) = AB + AC 5. Every element has a complement 0'= 1 1'= 0 A+A =1 A A = 0 and (A ) = A Note that each of these postulates contains a pair of rules; they are called duals of each other. In general, the dual of any Boolean expression is obtained by replacing all + s with s and all 0's with 1's. If a Boolean expression or equation is determined to be true, then its dual will also be true, and it is not necessary to prove the dual independently. This property of is called the Principal of Duality. By the way, it would seem that the functions AND and OR correspond in many ways with the arithmetic operators MULTIPLICATION and ADDITION, respectively. This is 8
9 certainly true regarding identities, commutativity, and associativity. Note, however, that the distributive rule is a bit different in that both Boolean operators are distributive across each other, while multiplication is distributive across addition, but not vice versa. Many additional theorems of can be proved using the postulates given above. Theorems, of course, can then be used to prove other theorems and results. The most important and useful theorems are presented here, with their proofs. In each case, as a result of the Principle of Duality, only one of the dual pair need be proved. Theorems 1. A+A = A AA = A 2. A+1 = 1 A0 = 0 3. A+AB = A A(A+B) = A Absorption 4. A+A B = A+B A(A +B) = AB Adsorption 5. (A+B) = A B (AB) = A +B De Morgan s Theorem The primary use for these theorems is for the simplification of Boolean expressions, generating canonical forms, and for proving the equivalence of Boolean expressions. Simplification and Equivalence a) Simplify AB+AB C = A(B+B C) Distributivity = A(B+C) Adsorption = AB+AC Distributivity b) Prove ABC+ ABC +AB C is equivalent to AB + AC ABC+ABC +AB C = AB(C+C ) +AB C Distributivity = AB+AB C Complement = AB+AC (proved in a) Canonical Forms The canonical Boolean forms are the Disjunctive Normal Form, referenced repeatedly 9
10 above, and the Conjunctive Normal form (CNF). Recall that the DNF is a sum-ofproducts form in which every minterm contains every variable used in the function; the CNF is a product-of-sums form in which every maxterm contains every variable used in the function. The Exclusive OR (XOR) The Exclusive Or (XOR) is a binary infix operator defined by the following truth table: A B f XOR That is, it is the same as the OR function with the exception that it is not TRUE when both A and B are TRUE. Thus, f XOR is TRUE whenever A is TRUE or B is TRUE, but not if both are TRUE. The XOR function can be expressed in terms of AND, OR and NOT as A XOR B = A B + AB [B8] as indicated by the truth table. From now on we will use the symbol for the XOR operator. Two other interpretations are of interest, and make this a very useful function: 1. The result is 1 when the arguments are different; the result is 0 when the arguments are the same. 2. The result is 1 when an odd number arguments are 1, and 0 when an even number of arguments (including none) are 1. Like the OR, the XOR is associative and commutative, but it is not distributive. A B = B A A (B C) = (A B) C Consider f = A B C. The truth table for this function is A B C f 10
11 This demonstrates the second of the two bullets above, since a result of 1 appears only if there are one or three 1's among the arguments. A useful consequence of this property of the XOR is that an XOR function can be complemented simply by complementing any odd number of variables. if f = A B C, f = A B C = A B C = A B C = A B C = (A B C) Another useful consequence of the this is that any Boolean expression can be complemented by XORing it with 1: f = f 1 11
12 Review Questions 1. Which of the following truth tables represent XOR (or NOT XOR) functions? a. b. c. d. A B C f A B C f A B C f A B C f Which of the following DNF expressions represent XOR (or NOT XOR) functions? a. AB + A B b. AB C + A B C + A BC c. ABC + A BC + AB C + ABC NAND and NOR The NAND and NOR functions are the invert (complement) of the AND and OR functions, respectively. That is, they have the following truth tables, in which all the 1's and 0's in the output columns for the AND and OR operators have been complemented (inverted) to get the NAND and NOR functions, respectively. A B A NAND B A B A NOR B Algebraically, A NAND B = NOT (A AND B) = (AB) A NOR B = NOT (A OR B) = (A+B) The importance of these two functions, or operators, lies in the fact that each of them is 12
13 sufficient to implement any Boolean Expression! This is valuable in computer chip manufacturing as it means a chip can be built containing only one kind of circuit and personalized only through metalization. The alternative, using ANDs, ORs and NOTs, would require chip designers to guess, before any engineering design has been done, what percentage of each kind of circuit might be required, and also what the placement of the circuits ought to be across the chip. We will use the symbol for the NOR function and for the NAND. We now show that the basic Boolean operators, AND, OR, and NOT can all be expressed using only NANDs or NORs. From this the assertion in the previous paragraph follows. Logic Circuits 1. A = (A1) = A 1 A = (A+0) = A 0 A = (AA) = AA A = (A+A) = AA 2. AB = ((AB) ) = (A B) = (A B) 1 = (A + B ) = ((A 0) + (B 0)) = (A 0)(B 0) 3. A+B = (A B ) = A B = (A 1)(B 1) = ((A+B) ) = (A B)0 4. A B = A B+AB = ((A 1)B)(A (B 1)) = (AB+A B ) = AB A B = ((A 0)(B 0))(A B) Logic circuits are electronic devices which implement Boolean expressions. Such circuits can be exceedingly complex, but they are built up from a few fundamental circuits, called logic gates, or just gates (no relation.) As you might expect, there is an AND gate, an OR gate, and a NOT gate (which is commonly called an inverter, since it inverts the level of the incoming signal.) Since these gates are commonly used to implement more complex logic circuits, there is a standardized set of figures which are used to draw such circuits. 13
14 The function f = ABC + D would be implemented in hardware as Practice Problems - Logic Circuits 1. Draw the logic circuits corresponding to each of the following Boolean expressions. Do not simplify the expressions. a. AB+C(A +B) c. (A+B )(BC +A)+D b. A+B+C(A +C ) d. AB +A B 14
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# Card game - Insidious casino
Here's the question: A casino offers a card game using a normal deck of 52 cards. The rule is that you turn over two cards each time. For each pair, if both are black, they go to the dealer's pile; if both are red, they go to your pile; if one black and one red, they are discarded. The process is repeated until you two go through all 52 cards. If you have more cards in your pile, you win $100; otherwise (including ties) you get nothing. The casino allows you to negotiate the price you want to pay for the game. How much would you be willing to pay to play this game? Here's the solution: This surely is an insidious casino. No matter how the cards are arranged, you and the dealer will always have the same number of cards in your piles. Why? Because each pair of discarded cards have one black card and one red card, so equal number of red and black cards are discarded. As a result, the number of red cards left for you and the number of black cards left for the dealer are always the same. The dealer always wins! So we should not pay anything to play the game. What I don't understand about the solution is this part: "As a result, the number of red cards left for you and the number of black cards left for the dealer are always the same". I know that for each pair of discarded cards, one black and one red cards are discarded. But how is this statement enough to conclude that 'the number of red cards left for you and the number of black cards left for the dealer are always the same'? Isn't this completely ignoring obtaining a pair of black cards or a pair of red cards??? ## 3 Answers So there are 26 red cards and 26 black cards. If when you are done, you have$p$pairs of mixed cards (1 red and 1 black) then then the total number of red and black left over (ie not in the mixed pairs) is$26-p$. This is the case for black and red. Number of pairs of reds and number of pairs of blacks is therefore$(26-p)/2$. It also allows you to deduce that$p$is going to be even. So counter offer to the casino. I'll pay if the number of mixed pairs is odd and you pay if it's even! If it helps to visualize, try playing the game with only 2 red and 2 black, then 3 red and 3 black and so on, and you can see that the piles will always be the same. Either 0 piles or 1 pile each with 4 cards total, either 0, 1, or 2 piles each with 6 cards total. So if you gain an advantage by making a pile of your own before your opponent, you have x cards of your color left in the deck, but the opponent has x + 2. Since a match requires one card from each of your color your opponent will always have +2 cards until they make a match of their own. However, the same logic applies to them, so they can never gain an advantage either. But still win the money when the piles end up the same. While the other answers have already explained that you literally have zero percent chance to win given the current rules, I figured it was worth checking how the odds would change if you were also given the option to discard a pair of cards before revealing them. In other words, when choosing to discard there is a chance that you will discard a pair of red or black. As a function of the (randomly) discarded pairs of cards, I simulated the chance to win (average of 1e7 runs per data point). If you were to always discard the first pair of cards and no others, then the odds would jump from 0 to just under 25% chance to win. The best odds (~39% chance to win) are achieved when approximately half of the deck is discarded. So interesting alternatives (while maintaining the house's advantage) are to offer \$25 and then discard the first pair independent of what it is or to split the deck and offer \\$40 on either of the decks.
• What a difference a discard makes! Exploring a puzzle around its edges, as here, adds extra sophistication (sometimes also art, as in this case) to Puzzling. – humn Sep 4 '17 at 14:02
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# Problem 48
Show 40 post(s) from this thread on one page
Page 1 of 2 12 Last
• Apr 10th 2008, 09:20 PM
ThePerfectHacker
Problem 48
1) Let $\displaystyle n\geq 2$ prove that $\displaystyle 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n}$ is not an integer.
• Apr 21st 2008, 01:09 PM
Math's-only-a-game
Quote:
Originally Posted by ThePerfectHacker
1) Let $\displaystyle n\geq 2$ prove that $\displaystyle 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n}$ is not an integer.
Let k ΞZ such that 2^k £ n < 2^k+1
Let m be the least common multiple of 1,2,3,…,n except 2^k.
Then multiplying S = 1 – 1/2 + 1/3 -…..± 1/n by m we have:
mS = m – m/2 + m/3 -……± m/n
Each number on the right hand side is an integer except m/2^k and hence Sm is not an integer, which implies Sm is not an integer. (Hi)
• Apr 23rd 2008, 02:14 PM
icemanfan
By the alternating series theorem, the partial sum will always be less than one but greater than zero, and therefore not an integer.
• Apr 25th 2008, 12:31 PM
Aryth
Quote:
Originally Posted by ThePerfectHacker
1) Let $\displaystyle n\geq 2$ prove that $\displaystyle 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n}$ is not an integer.
The series you presented is the Alternating Harmonic Series, which is Conditionally Convergent, the series is represented by:
$\displaystyle \sum_{n=1}^{\infty} \left(\frac{(-1)^{n+1}}{n}\right)$
The series' terms look like such:
$\displaystyle 1 - \frac{1}{2} + \frac{1}{3} - ... \pm \frac{1}{n}$
This series converges to $\displaystyle \ln{2}$
Since the series converges to $\displaystyle \ln{2}$ and since:
$\displaystyle |a_{n+1}| < |a_n|$
Then for $\displaystyle n \geq 2$ the series can never reach one since it is incrementing up or down by smaller amounts. Since you subtract $\displaystyle \frac{1}{2}$ from 1 for n=2, and since the terms are decreasing and alternating in sign, then the series will never reach one again, therefore, this can't be an integer for $\displaystyle n \geq 2$ because all terms are decreasing,therefore the partial sums remain between 1 and 0.
• Apr 25th 2008, 12:36 PM
1 +1/2 + 1/3 + 1/4 + .... ===>A
and
1/2 + 1/4 + 1/6 +.... =====>B
to get the required series:
A - 2B
• Apr 25th 2008, 01:14 PM
Aryth
Yeah, that is a distinct possibility...
The Alternating Series does equal:
H(n) - H(2n)
Where H(n) is the n-th harmonic number
• May 14th 2008, 06:11 AM
Henderson
Huh. I stayed away from this one because I didn't pick up on the series alternating- I read $\displaystyle \pm \frac{1}{n}$ as saying each term could either be added or subtracted, without nessecarily alternating.
Is there a similar solution to this problem?
• Jun 4th 2008, 10:01 AM
Aryth
You can't know what sign the last number of the series is going to be, that all depends on n, so the $\displaystyle \pm$ means that it can be positive or negative depending on n. The initial pattern reveals an alternating series.
• Jun 6th 2008, 12:14 PM
Jacobsen
[FONT='Cambria Math','serif']My first thought was to try an inductive argument, but I had a lot of difficulty getting it going. I dont think what I came up with is sound, but nevertheless I decided to post what I came up with.
Proof. It suffices to show that for all
n≥2; 1-1/2+1/3-
±1/n ∈ (0,1).
Let Pn denote the proposition that
1-1/2+1/3-
±1/(n-1) ∈ (0,1)
and
1-1/2+1/3-
±1/(n-1)±1/n∈ (0,1).
Then P3 is true since
1-1/2=1/2∈ (0,1)
and
1-1/2+1/3=5/6∈ (0,1)
Assume Pn is true and that n is even. Then
1-1/2+1/3-
+1/(n-1) ∈ (0,1)
and
1-1/2+1/3-
+1/(n-1)-1/n ∈ (0,1).
Because 1/(n+1) < 1/n, it follows from the inductive hypothesis that
1-1/2+1/3-
+1/(n-1)-1/n+1/(n+1) ∈ (0,1).
The case where n is odd is similar. So by the principle of mathematical induction, for all n ≥ 3, Pn is true and hence for all n ≥ 2, 1-1/2 +1/3 -
±1/n ∈ (0,1) and hence not an integer. //
[/FONT]
• Jul 7th 2008, 03:14 PM
meymathis
Quote:
1 +1/2 + 1/3 + 1/4 + .... ===>A
and
1/2 + 1/4 + 1/6 +.... =====>B
to get the required series:
A - 2B
These two series do not converge.
• Jul 7th 2008, 08:06 PM
meymathis
Let $\displaystyle A_N = 1 - \frac{1}{2} + \ldots \pm\frac{1}{N}$ which is just the partial sums.
Consider the (sub) sequence of partial sums:
$\displaystyle O_N = 1 - \frac{1}{2} + \ldots + \frac{1}{2N+1}$ for $\displaystyle N\geq1$
$\displaystyle O_N$ is a subsequence of $\displaystyle A_N$ which as noted above converges to ln(2) (derive using MacLauren expansion of ln at x=1). Then $\displaystyle O_N\rightarrow\ln(2)$.
$\displaystyle O_N$ is monotonically decreasing:
$\displaystyle O_{N+1}-O_N = - \frac{1}{2N+2} + \frac{1}{2N+3} < 0$
Note that $\displaystyle O_0=1$ and so $\displaystyle 1>O_N\geq\ln(2)\approx0.693$ for $\displaystyle N>0$ and so cannot be an integer.
Likewise for the partial sums:
$\displaystyle E_N = 1 + \ldots - \frac{1}{2N}$ for $\displaystyle N\geq1$
except that $\displaystyle E_N$ monotonically increases from 1/2 to ln(2).
Put it together and we just showed the odd and even elements of the partial sums $\displaystyle A_N$ are never integers after 1.
• Jul 30th 2008, 05:58 AM
Lore
Quote:
1 +1/2 + 1/3 + 1/4 + .... ===>A
and
1/2 + 1/4 + 1/6 +.... =====>B
to get the required series:
A - 2B
A - 2B = 0? Considering B = A/2...
• Sep 8th 2008, 09:59 AM
bkarpuz
Quote:
Originally Posted by Lore
A - 2B = 0? Considering B = A/2...
As meymathis said, these two series do not converge, in other words, $\displaystyle A=\infty$ and $\displaystyle B=\infty$.
So you do algebric operations on infinite numbers, which may confuse your mind.
• Oct 10th 2008, 10:49 PM
Suzan
Quote:
Originally Posted by ThePerfectHacker
1) Let $\displaystyle n\geq 2$ prove that $\displaystyle 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n}$ is not an integer.
Suppose that http://plus.maths.org/MI/plus/issue1...s/img-0001.png. Choose an integer http://plus.maths.org/MI/plus/issue1...s/img-0002.png such that http://plus.maths.org/MI/plus/issue1...s/img-0003.png.
Then http://plus.maths.org/MI/plus/issue1...s/img-0004.png
Consider the lowest common multiple of http://plus.maths.org/MI/plus/issue1...s/img-0005.png. This number will be of the form http://plus.maths.org/MI/plus/issue1...s/img-0006.png, where http://plus.maths.org/MI/plus/issue1...s/img-0007.png is an odd integer. Now multiply both sides of the equation by this number, to get
http://plus.maths.org/MI/plus/issue1...s/img-0008.png
Now, when multiplied out, all the terms on the left will be integers, except one:
http://plus.maths.org/MI/plus/issue1...s/img-0009.png
is not an integer, since http://plus.maths.org/MI/plus/issue1...s/img-0007.png is odd. So the left hand side is not an integer, and hence neither is the right hand side. That means that http://plus.maths.org/MI/plus/issue1...s/img-0010.png is not an integer.
Not:
http://plus.maths.org/issue12/features/harmonic/index.html
• Oct 12th 2008, 07:48 AM
shawsend
Quote:
Originally Posted by ThePerfectHacker
1) Let $\displaystyle n\geq 2$ prove that $\displaystyle 1 - \frac{1}{2}+\frac{1}{3} - ... \pm \frac{1}{n}$ is not an integer.
I'm confussed since $\displaystyle \sum_{k=1}^{n}\frac{(-1)^{n+1}}{k}<1$
However $\displaystyle H_n=\sum_{k=1}^{n}\frac{1}{k}\to\infty$ and my understanding is that $\displaystyle H_n$ is never an integer for $\displaystyle n>1$. This one would seem to be more interesting to prove.
I think that's what Susan did. Never mind but perhaps we should make it explicit that's what's going on.
Show 40 post(s) from this thread on one page
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# Iowa Probability and Statistics (IowaCore) Practice
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Interpreting Categorical and Quantitative Data 33% Making Inferences and Justifying Conclusions 34% Conditional Probability and the Rules of Probability 33%
• Questions: 1,313
• Technology Enhanced Items: 41
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• Vocabulary Terms: 122
### Test Standards
Interpreting Categorical and Quantitative Data 1. (S-ID.1) Data Plots 2. (S-ID.2) Data Distribution 3. (S-ID.3) Interpret Differences 4. (S-ID.4) Normal Distribution 5. (S-ID.5) Categorical Data 6. (S-ID.6) Scatter Plot 7. (S-ID.6a) Fit Function 8. (S-ID.6b) Assess Fit 9. (S-ID.6c) Fit Linear Function 10. (S-ID.7) Slope And Intercept 11. (S-ID.8) Correlation Coefficient 12. (S-ID.9) Correlation And Causation Making Inferences and Justifying Conclusions 1. (S-IC.1) Understand Statistics 2. (S-IC.2) Data-Generating Process 3. (S-IC.3) Purposes And Differences 4. (S-IC.4) Estimate Mean 5. (S-IC.5) Randomized Experiment 6. (S-IC.6) Evaluate Reports Conditional Probability and the Rules of Probability 1. (S-CP.1) Set Of Outcomes 2. (S-CP.2) Independent Events 3. (S-CP.3) Conditional Probability 4. (S-CP.4) Frequency Tables 5. (S-CP.5) Probability And Independence 6. (S-CP.6) Find Probability 7. (S-CP.7) Addition Rule 8. (S-CP.8) Multiplication Rule 9. (S-CP.9) Permutations
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# Multiplication
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• Students fill in 40 missing products on a 10x10 multiplication grid.
• Students fill in 80 missing products on a 10x10 multiplication grid.
• Students fill in 48 missing products on a 9x9 multiplication grid.
• Students fill in 10 missing products on a 5x5 multiplication grid.
• Students fill in 64 missing products on a 9x9 multiplication grid.
• A one page lesson on the Properties of Addition & Multiplication includes: commutative property, associative property, distributive property, identity property, and zero property. It is followed by two pages of practice equations. Common Core: Math: 3.0A.5 3.0A.6, 4.OA.1
• A one page math lesson on the Properties of Equality includes: Addition Property of Equality, Multiplication Property of Equality, Reflexive Property of Equality, Symmetric Property of Equality, and the Transitive Property of Equality. It is followed by two pages of equations for practice.
• A page of rules and a page of practice for scientific notation, including; multiplication, numbers, with an answer sheet.
• Factoring practice worksheet - Students will write all the factors for a particular number.
• Four pages of decimal multiplication with twenty equations per page. Products include 1-5 decimal places.
• 6 pages of worksheets to practice multiplication to 12x12, with answer sheets.
• Jacob loves school, but he hates multiplication. A realistic fiction reading comprehension.
• One page with nine illustrated black and white bunny-themed multiplication flashcards (x1)
• Create a story problem for this answer: Jacob had four caramel apples left over. Six word problems.
• Amanda had \$3.00. She bought a hot dog for \$1.35, chips for 35 cents, and a drink for 85 cents. Did she have enough money? Did she have money left over? If so, how much? Six word problems.
• Joseph has two cousins. The sum of their ages is 18 years. One cousin is four years older than the other. Mrs. Smith guessed that the ages were 7 and 11. Was her guess correct? Six word problems.
• Heather says, "I have two numbers in mind. When I subtract the smaller from the larger, the difference is seven. When I multiply the two numbers, the product is eighteen. What are my two numbers?" Six word problems.
• Great for practice. Use alone, or link all the bookmarks together on a ring.
• "Mary did six exercises and studied three pages a day. By the end of one week, how many exercises had she done, and how many pages had she studied?" One page of word problems, with addition, subtraction, and multiplication.
• Four pages of worksheets for practicing multiplication by 0, 1, and 2. With answers and cute illustrations.
• 5 pages of worksheets to practice 1-digit multiplication.
• 5 pages of worksheets to practice 2-digit multiplication, plus answers.
• 5 pages of worksheets to practice single digit multiplication, plus answers.
• This lesson is designed to help students practice multiplication skills. Combine groups of threes and fives with ones to complete a chart; includes teaching suggestions and answer sheets.
• 5 pages of worksheets to practice multiplication up to 10, plus answers.
• Using teacher-created dice, students use this simple game to practice multiplication with factors between four and nine.
• A blackline house for multiplication and division "fact families" to live in.
• Students write multiplication problems (with sums up to 40). When the answer to the problem is called, they cover the square. A fun variation on a popular favorite.
• Worksheets to practice 3-digit multiplication. 5 pages plus answers.
• By 10, by 9, by 5, by 3, and by chunks. This unit contains tricks for multiplication and for checking your work. A playful (and very useful) approach to multiplication. This unit is presented at three levels of varying complexity.
• "When you read PRODUCT... multiply!" Five posters of guidelines to help with reading word problems as equations.
• Master the multiplication tables with these wacky flashcard-holding animals (a different animal for each set). Up to 12 x 12.
• Worksheets for practicing multiplication by 10, 11 and 12.
• Book comprehension and vocabulary enhancement for this installment of Marc Brown's popular "Arthur" series. Arthur has trouble with truth in advertising.
• These fish-themed pages serve as a colorful guide for word problems, displaying the mathematical symbol (x) that accompany the most common Multiplication Keywords.
• Math circles with numbers already inserted to practice all facts from 1 to 10. Students fill in the answers. Can be used for either addition or multiplication.
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| 3.8125 | 4 |
CC-MAIN-2017-22
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http://www.thecalculator.co/others/Bulk-Modulus-Calculator-649.html
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| 746,214,522 | 8,190 |
This bulk modulus calculator helps you compute the bulk elasticity of a material or object based on volume and pressure change in various units. You can discover the formula used and more on the subject below the form.
Instruction: Please input four fields from the five below!
Initial Volume:*
Final Volume:*
Initial Pressure:*
Final Pressure:*
Bulk Modulus:*
## How does this bulk modulus calculator work?
This is a useful tool that allows you to calculate the bulk elastic properties of a material through its initial and final volume and the pressure change according to initial and final pressure. Whilst the main function is that of calculating this, the bulk modulus calculator basically allows you to input any four of the equation elements in order to be delivered the fifth one in the result.
There are several measurement units you can use for the volume, from the recommended SI in cube metres (m3) to cube foot (ft3) or liters (l). When adding pressure you are asked to choose from Pa, bar, atm and mmHg.
This is the formula used:
Bulk modulus = -Vi * (Pf –Pi) / (Vf- Vi)
where
Vi = initial volume in m3
Vf = final volume in m3
Pi = initial pressure in Pa
Pf = final pressure in Pa
Bm = bulk modulus in Pa
## Example calculation
Let’s take the case of an object with a volume Vi = 5 m3 being reduced to a volume of Vf = 3.4 m3 by a change in pressure of 7 Pa ( Pi = 1.6, Pf = 8.6).
Bulk modulus = -5 * (8.6 –1.6) / (3.4 - 5)
Bulk modulus = 21.875 Pa
## What is bulk modulus?
This is a measure/ ratio that defines how much a material (solid or fluid) will compress under a given amount of change in applied external pressure, therefore it describes the volume elasticity of the material. The reciprocal would be the compressibility of the material. The standard unit is the pascal (Pa) but newton per square metre (N/m2) is often used. Talking about elasticity, it also means that after the pressure is removed, the material regains its volume. Bulk modulus can also be described as the pressure divided by the strain or relative deformation.
What is interesting about this ratio is that it can also describe the amount of energy the Earth’s crust has stored and thus it is used in the study of earthquakes. Did you know that glass has a bulk modulus value of 35 to 55 GPa or that diamonds exceed it with a value of 443 GPa?
28 Apr, 2015 | 0 comments
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http://manualzz.com/doc/37899/laboratory-manual-for-electric-energy-engineering-ee
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| 201,295,514 | 18,415 |
### Laboratory Manual For Electric Energy Engineering EE
```Laboratory Manual
For
Electric Energy Engineering
EE-360
Electrical Engineering Department
King Fahd University of Petroleum & Minerals
Dhahran, Saudi Arabia
Experiment
Title
No.
1
Three Phase Circuit
Page
No.
3
2
Three Phase Power Measurement
6
3
Magnetic Circuit
9
4
Equivalent Circuit of Transformer
12
5
Regulation and efficiency of a single phase
Transformer
16
6
Load Characteristic of shunt and compound
DC generator
19
7
Torque Speed Characteristic of DC shunt
and compound motors
22
8
Determination of Parameters of
Synchronous Generators
26
9
Torque Speed Characteristics of 3Φ
Induction Motors
29
10
Determination of Induction Motor Parameters
32
2
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
Electric Engineering Department
EE 306
Electric Energy Engineering - Experiment#1
THREE PHASE CIRCUITS
Objectives:
•
•
•
To learn how to make wye (Y) and delta (∆) connections
To study the relationship between voltage and current in three phase
circuits.
To make power calculations.
Apparatus:
•
•
•
•
2 AC voltmeters
2 AC Ammeters
1 3Φ variable AC power supply
Theory :
In a Y connection , the line and the phase quantities are related by:
Vp=VL/√3
(1)
Ip=IL
(2)
Whereas the relationships for a delta connection are
Ip=IL/√3
(3)
Vp=VL
(4)
The real and reactive powers for a 3 Φ circuit (either Y or ∆ connection) are
given as
3
P=√3 VL IL cos θ
(5)
Q=√3 VL IL sin θ
(6)
Where θ is the power factor angle of the balanced load
Procedure:
A:
Y – Connection
a
A
A
V
V
3 Phae Ac
B
b
C
c
Fig. 1 :
The Y - Connection
2. Switch the load to unity power factor mode
3. Select the balanced load from each phase
4. With the load switch off turn the power supply on and adjust the line to
neutral voltage to 120 volt or VL = 208 volt
5. Measure the line and phase voltages and currents. Make the table similar to
table1 on a separate page and enter your readings in the first 4 columns
VL
Vp
IL
Ip
VL / Vp
IL
Ip
/ P
4
Q
Remarks
Take three readings, one at the rated value of the load current (8A), one at ½
rated load and one at ¼ rated.
6. Repeat step 5 for 0.8 and 0.8 leading power factor loads
B:
∆ Connection
1. Connect the three phase load as shown in fig. 2
A
A
a
3 Phae ac
N
V
N
b
N
B
A
C
c
Fig. 2 : The Delta- Connection
2. Turn the power supply on and adjust for 120V A.C (Note: Vp=VL for ∆)
3. Repeat step 5 of the Y connection for unity, 0.8 lagging and 0.8 leading
power factors and enter in a table similar to table 1, call it table 2.
Report
1. Complete tables 1 and 2.
2. Calculate the total real and reactive powers.
3. Draw phasor diagrams showing the line and phase voltages and currents for
both Y and ∆ connections. Draw only for rated load, unity power factor
condition.
4. Verify the relationships for the phase and the line voltages and currents and
state reasons for any errors.
5
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
Electrical Engineering Department
EE 360
Electric Energy Engineering -
Experiment # 2
THREE PHASE POWER MEASUREMENT
Objectives:
1. Measure power in balanced Y and ∆ systems.
2. Determine power factor of 3Φ systems.
Apparatus:
2 Wattmeters
1 Voltmeter
1 Ammeter
1 3Φ variable AC power supply (Variac)
Theory:
a
M
A
V1
A
Three
Phase
V
B
b
V1
c
C
M
Fig.1: Two Wattmeter Connection
If two wattmeters are connected to measure the power of any 3Φ load, it can be
shown that the wattmeters will read V1
6
P1 = VL IL cos ( 30 – θ )
P2 = VL IL cos ( 30 + θ )
(1)
(2)
Where θ the power factor angle of the load. From (1) and (2) we can show that
the total power
PT = P1 + P2 = 3 VL IL cosθ
(3)
tanθ = 3 ( P1 - P2 ) / ( P1 + P2 )
(4)
Procedure
1. Connect the circuit as shown in fig 1. Connect the
2. Before you switch on, have your connections cheeked by the instructor.
3. Set the supply voltage to 200 V from a variac
4. Select the load power factor to be unity
5. On a separate sheet of paper make a table with 11 columns as shown in
table.1.
Pf
P1
P2
VAB
VCB
IA
PT
(Watt) (Watt) (Watt) (Volt) (Volt) (amp)
Pf
calc.
Pf
Error
(%)
PT
Calc.
Power
Error
calc.
Table.1: Results for Y connection
8 A, one for ½ rated and one for ¼ rated loads.
7. Repeat step 6 for 0.8 lagging as well as leading power factor conditions.
8. Connect the three phase load in ∆.
9. Set the supply voltage to 100 volts (VL= VP for ∆).
10. Repeat step 6 for unity, 0.8 lagging and 0.8 leading power factor conditions.
Enter the results in a table similar to table 1.call table 2.
7
Note: At a certain power factor, one of the wattmeters may try to read
backwards. Switch the supply off, reverse the voltage OR the current coil
connection. Mark the reading as negative.
Report
1. Using the wattmeter readings, compute the power factor from equation (4).
Enter it as pf (calculated) in tables.1 and 2. Calculate the percent error
between the calculated and the recorded power factors.
2. Use equations (1) and (2) to calculate the total power. Compare it to the
measured total power and enter the percent error in the tables.
3. Comment on the levels of error between the computed and measured values.
State any sources of error.
4. Draw a phasor diagram and show why equations (1) and (2) can be used to
calculate the total power.
8
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
Electric Engineering Department
EE 306
Electric Energy Engineering - Experiment#3
MAGNETIC CIRCUITS
Objective:
1. To determine the B-H characteristics of an iron core
2. To find the relative permeability (µr)
3. To calculate the reluctance “R”
Apparatus:
1 Rectangular laminated core
1 coil
1 voltmeter
1 ammeter
1 variable AC supply
Theory:
I
A
N
V
Fig. 1 : A simple rectangular core
If a current of 1 A, flows from a supply of E volts through a coil of N turns, as
shown in fig 1, the magnetic field intensity can be written as
H = NL / LC
9
(1)
From faraday’s law of electromagnetic induction, the rms values of the induced
voltage across the coil (E) is
E = ωNΦ
= ωNAB
(2)
B=µH
(3)
From (1), (2) and (3) it is clear that E-I characteristic of the core is equivalent to
the B-H characteristic. Further, it can be shown that
E = ωN2A µ I
Lc
(4)
Where, the permeability can be written as:
µ = µr µo;
µo = 4 π x 10-7
(H/n)
The reluctance of the core can be expressed as:
R= NI / Φ
= Lc / (µA)
(5)
Procedure
1. Find the typical dimensions of the core. The instructor may help you to get
the accurate numbers.
2. Connect the circuit as in fig 1
3. On a separate sheet of paper make a table as shown below:
Table 1
E
I
K= E / I
µr
R
4. Set the input voltage of 10V. Record the current and enter them in table 1.
5. Repeat step 4 up to 150 volts in steps of 10 volts.
10
Report
1. Plot E Vs I on a graph paper.
2. Find K, and R for each reading and complete the table. Here,
K=E/I
µ r=
KLc
2 π fN2A µo
3. Plot µ and R as functions of I
4. Derive equations (4) and (5)
Core Dimensions:
Lc = 40 cms
N = 400 turns
A = 9 Sq. cms
11
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
Electric Engineering Department
EE 306
Electric Energy Engineering - Experiment#4
EQUIVALENT CIRCUIT OF TRANSFORMER
Objectives:
1. To determine the equivalent circuit of a single phase transformer
2. To verify the voltage current relationship
Apparatus:
1 Single-phase transformer
1 Variable AC power supply
1 AC voltmeter
2 AC ammeters
1 Wattmeter
Theory
The approximate equivalent circuit of a transformer is given in Fig. 1.
Req
Xeq
Xm
Rc
Fig 1. Equivalent Ciruict of transformer
Where, Rc =1/g and xm =1/b. These quantities are obtained from the open circuit
power, voltage and current measurements. These are
12
and,
Y = g - jb = Io / Vo
(1)
g = Po / Vo2
(2)
b = √ |Y|2 – g2
(3)
The equivalent resistances and reactances (Req, Xeq) are obtained from the
current, voltage and power measurements in the primary winding when the
secondary is shorted. These are written as
Req = Psc / I2sc
|Zeq| = Vsc / Isc
Xeq = √|Zeq|2 - Req2
(4)
(5)
(6)
Procedure
1. Note the current, voltage and volt-ampere ratings of both windings of the
transformer. Note the turns ratio
2. Connect the circuit as shown in Fig2. with the high voltage side open
circuited
3. Adjust the supply voltage until the voltage on the primary side is the rated
value.
4. Record the current, voltage and power in this condition. Take another reading
at 110 % of the rated value.
5. Next, connect the transformer for the short circuit test as given in Fig 3. The
variable supply will be on the high voltage side.
13
L
Variable AC
Source
Digital
Wattmeter
220V side
Open Circuit
110 / 220 V
N
Fig. 2 : The Open Circuit Test connection
6. Gradually increase the supply voltage from zero until the rated current flows
in the shorted secondary winding
7. Record the current, voltage and power. Repeat step 6 for 110 % of rated
current and record the values.
A
L
Variable AC
Source
Digital
Wattmeter
110V side
short Circuit
220 / 110 V
N
Fig. 3 : The Short Circuit Test connection
8. Connect the circuit as shown in Fig. 4 for a load test
9. Adjust the supply voltage and the resistive load such that rated current flows
through the load at rated voltage
Measure the voltages and currents on both sides of the transformer
14
A
L
Variable AC
Source
Digital
Wattmeter
V
220 / 110 V
N
Fig. 4 : The Load Test connection
Report
1. Calculate Rc, Xm, Req and Xeq from the open circuit and short circuit tests.
2. Draw the approximate equivalent circuit diagrams and label the parameter
values. Note that some of the values have to be transferred to the other side of
the winding by multiplying with approximate constant.
3. For the unity power factor loading condition of Fig 4, calculate the primary
current and voltage using the equivalent circuit you obtained. Start with the
measured values of current and voltage on the load side.
4. Compare the calculated quantities with measured ones and compute the
percent error
5. State the possible sources of errors, if any.
15
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
Electric Engineering Department
EE 306
Electric Energy Engineering - Experiment#5
REGULATION AND EFFICIENCY OF A SINGLE PHASE
TRANSFORMER
Objectives:
1. To determine the regulation of a transformer
2. To determine the efficiency of a transformer
Apparatus:
1 Single-phase transformer
1 Variable AC power supply
2 Voltmeters
2 Ammeters
2 Wattcmeters
Theory
The voltage regulation of transformer at rated load is defined as:
VR = (Vno load - Vrated) / Vrated
(1)
If the approximate equivalent circuit of a transformer is used then for a lagging
V1= Vno load = Vrated <0o + I (cos θ – j sin θ) (Req + j Xeq)
= Vrated <0o + (Req cos θ + I Xeq sin θ) + j (- I Req sin θ + I Xeq cos θ)
Neglecting the imaginary part on the right hand side,
16
(2)
VR = I (Req cos θ + Xeq sin θ)
Vrated
(3)
The efficiency of the transformer can be written as
η = Power output / Power input
(4)
Or
η =
Power Output___
Power output + Loses
The losses are,
Core loss = No load power input – No load copper loss
Copper loss = I22 Req
Procedure :
L
Variable AC
Source
Digital
Wattmeter
Digital
Wattmeter
220 / 110 V
N
Fig. 1 : A Transformer with Load
1. Record the ratings of the transformer
2. Note down the parameters of the approximate equivalent circuit from the
previous experiment. If you are using a different transformer, perform the
open circuit and short circuit test again.
3. Connect the circuit as shown in Fig.1.
4. Make a table on the separate page as table.1.
5. Select unity power factor load.
6. Adjust the input voltage so that the load voltage is the rated value for a
certain load current. Record Pi, Po, V2 and I2. Switch the load off and record
V2. This is V2 (no load)
7. Repeat step 6 for various loads until you have reached the rated current. Take
17
8. Select 0.8-power factor lag. Repeat step 6 for rated current
9. Repeat step 8 for 0.8 p.f . leading.
Table 1
P.f
V2
I2
Pi
Po
V2
(No
η=
P2/Pi
VR
Η
(cal)
VR
from
eq3
Report
1. Calculate efficiency and voltage regulation fro your test results. Enter them in
columns 7 and 8 in table 1
2. Plot efficiency as function of load current for the unity power factor load
3. For rated, ½ and ¼ rated load, Calculate the efficiency from the equivalent
circuit. Enter them in table 1. Compare with measured values
4. Calculate the voltage regulation for rated load at unity, 0.8 lagging and 0.8
leading power factors using equation 3. Enter them in the table. Compare
5. State reasons of any discrepancy between the measured and the calculated
values
18
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
Electric Engineering Department
EE 306
Electric Energy Engineering - Experiment#6
Load Characteristics of Shunt and Compound DC Generators
Objectives:
1. To study the load voltage vs. current characteristics of shunt connected DC
generator.
2. Study the load characteristics of a compound generator.
Apparatus:
1 DC motor-generator set
1 Tachometer
1 DC Voltmeter
2 DC Ammeters
1 Power Supply
Theory:
The terminal voltage of a shunt generator is written as:
Vt = Ea – Ia Ra
(1)
Where Ia = If + IL
If is the shunt current and
For a short shunt compound generator, the terminal equation is modified to
Vt = Ea – Ia Ra - IL Rsc
(2)
Where Rsc is the resistance of the series winding.
19
Procedure:
1. Record the rated currents and voltages of the DC generator and the motor.
Record the rated speed of the motor and generator.
2. Make the connection as shown in fig.1.
+
R
+
LINE
If
-
rheostat
rheostat
+
+
+
M
-
DC
SUPPLY
A
Ia
+
G
-
Ea
SHUNT
Vt
V
-
SHUNT
+
-
A
-
Fig.1: Connection Diagram For Shunt Generator
3. Set the generator shunt field rheostat to its maximum value.
4. Set the motor shunt field to its minimum value.
5. Adjust the motor speed to almost rated value. You can go slightly higher than
the rated one. The motor speed can be adjusted by changing the resistance in
the motor field winding or with series resistance RLine.
6. Adjust the generator voltage to its rated value by controlling the field
rheostat. Keep the load disconnected during the voltage buildup.
Maintain the motor speed to same value.
8. Record the speed of the motor. Enter the load voltage, load current and field
current as in table.1 for different loading conditions. Take at least 10 sets of
Table.1
20
VL
IL
If
9. Repeat the procedure for the compound generator given in fig.2.
+
A
+
R
LINE
SERIES
Ia
+
M
-
DC
SUPPLY
+
G
-
IL
If
+
Ea
V
-
SHUNT
+
-
A
-
Fig.2: Connection Diagram for Compound Generator
REPORT:
1. Plot the load voltage and field current of the shunt generator against the load
current.
2. Repeat the above for the compound machine.
3. Find the voltage regulation at rated load from your experimental results for
both shunt and compound machines.
4. Comment which generator is better in terms of load characteristics and why?
21
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
Electrical Engineering Department
EE 306
Electric Energy Engineering - Experiment#7
Torque Speed Characteristics of DC Shunt and Compound Motors
Objectives:
1. To study the variation of speed of shunt motor when load is changed.
2. To study speed vs. load characteristics of a compound motor.
Apparatus:
1 DC motor- generator set
1 Tachometer
1 DC Voltmeter
2 DC Ammeters
1 Power supply
1 Resistance
Theory:
For DC shunt and long shunt compound motors, current and flux are related by:
Vt = Ea + Ia Ra
(1)
Ea = Ka ω m Φ
(2)
Which gives
ωm =
Vt − I a Ra
KaΦ
(3)
Using the equation
Ia = Tdev / (KaΦ)
(4)
We can write
22
ωm =
Ra
1
Vt −
Tdev
KaΦ
( K a Φ) 2
(5)
Equation (5) shows the relation between torque, speed, terminal voltage and flux
of the motor.
Procedure:
1. Record the rated voltage, current and speed of the motor and the generator.
The generator is used to load the motor.
2. Connect the circuit as shown in fig.1
A
A
+
DC
SUPPLY
Ia
+
M
-
+
G
-
Ea
+
DC FIELD
SUPPLY
-
V
-
Fig.1: The Shunt Motor Generator Connection
3. Adjust the generator field resistance to maximum and motor field to
minimum.
4. Start the motor and bring the speed to slightly more than rated.
5. Apply the generator field and buildup the voltage to its rated value.
switching in the load rack. Adjust the generator terminal voltage to the rated
value every time by varying the field rheostat and/or the field supply voltage.
7. Record the motor speed n (rpm) and the motor armature current Ia for every
8. Make connection as given in fig.2 for the compound motor.
23
A
A
+
DC
SUPPLY
SERIES
FIELD
+
M
-
+
DC FIELD
SUPPLY
-
+
G
-
V
-
Fig.2: The Compound Motor Generator Connection
9. Repeat steps 3 thru 7 for the compound motor.
Report:
1. Plot the speed vs. motor armature current for the DC shunt motor.
2. Repeat 1 for the compound motor.
3. Calculate the speed regulation from no load to full load of the DC shunt
motor.
4. Repeat 3 for the compound motor.
Compare the torque-speed characteristics of the two motors and note your
observation.
24
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
Electric Engineering Department
EE 306
Electric Energy Engineering - Experiment#8
Determination of Parameters of Synchronous Generators
Objectives:
1. To determine the synchronous impedance of an alternator.
2. To determine its voltage regulation.
Apparatus
1 3Φ alternator
1 DC motor
1 AC Voltmeter
1 DC Ammeter
1 DC voltmeter
1 DC power supplies
1 Tachometer
Theory:
For a certain excitation the synchronous impedance per phase of a synchronous
machine can be calculated as
Zs = Ea / Ia
(1)
Where Ea is the open circuit voltage per phase and Ia is the short circuit current.
The synchronous reactance then can be calculated as
(2)
X s = Z s2 − Ra2
Ra is considered as 1.5 times the armature DC resistance Rdc .Xs is the saturated
reactance when Ea is taken from the open circuit characteristics and
Ia is the corresponding short circuit current for the same excitation.
For a certain load current Ia, the internal voltage per phase can be written as
25
Ea = Vt + Ia ( Rs + jXs )
(3)
Where Vt is the terminal voltage per phase. Note, Ia is a complex number
The voltage regulation of the generator at the rated load is given as:
VR = (VNL-VFL)/VFL X 100%
(4)
Where, VNL = Ea
and
VFL = Vt (rated)
Procedure:
1. Note the rated values of current, voltage and speed of the synchronous
generator as well as the motor that will drive the generator.
2. Connect the motor generator set as shown in fig.1 for the open circuit test.
A
+
+
DC FIELD
SUPPLY
-
FIELD
DC
SUPPLY
E
A
-
C
DC MOTOR
B
SYN. ALTERNATOR
Fig.1: The Open Circuit Test
3. Adjust the alternator field rheostat to the maximum value and that for the
motor to the minimum value.
4. Adjust the motor speed to the synchronous speed of the alternator. You can
control the speed by the resistors in the line or in the motor field circuit.
5. Vary the field current in steps by varying the rheostat in the field circuit
and/or the supply voltage. Record the line-to-line voltage (E) and the filed
current If. Make sure that the speed remains constant through the whole test.
6. Take the readings upto 110 % of the rated voltage of the alternator.
7. Stop the motor and connect as in fig .2 for the short circuit test of the
alternator
26
IA
A
A
A
+
DC FIELD
SUPPLY
-
FIELD
DC MOTOR
C
B
DC MOTOR
SYN. ALTERNATOR
Fig.2 The Short Circuit Test
8. With the generator exciter off, bring DC motor upto synchronous speed.
Close the 3Φ switch and gradually increase the excitation. Record the field
current If and the armature current Ia. Take readings upto 120 % of the rated
generator current.
9. Switch the alternator exciter off. Stop the motor and make connection as
given in fig.3 for measurement of DC resistance of the armature.
B
A
+
+
DC
POWER
SUPPLY
V
-
-
C
A
Fig.3: DC Resistance measurement Of The Alternator
10. Adjust the DC power supply so that the current flowing through the alternator
winding does not exceed the rated value. The DC resistance is given as
Rdc = Vdc / 2Id c
The armature resistance Ra can be considered to be 1.5 times Rdc
Note: the armature DC resistance can also be measured by an accurate
millimeter, or by some resistance measurement bridge.
27
Report:
1. Using the OCC and SCC test results, plot EA and IA against If on the same
graph paper.
2. From the plotted graphs, determine Zs and Xs using equations (1) and (2).
Calculate only the saturated value.
3. Calculate, analytically, the voltage regulation of the generator for the
One. Rated load, unity power factor
Two. Rated load, 0.8 lagging p.f
Three.
Use equations (3) and (4).
28
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
Electric Engineering Department
EE 306
Electric Energy Engineering - Experiment#9
Torque Speed Characteristics of 3Φ Induction Motors
Objectives:
1. To determine the torque speed characteristics.
2. To determine slip-torque characteristics.
3. To observe variation of efficiency.
APPARATUS:
1 3Φ induction motor
1 Prony brake
2 Wattmeters
1 3Φ variable power supply
1 Tachometer
1 Single pole switch
1 Digital Torquemeter
Theory:
The slip of an induction motor is defined as
s=
ns − nr
ns
where
ns is the synchronous speed
nr is the rotor speed
The efficiency of the motor is calculated from the ratio of the output mechanical
power to input electrical power as
η=
Pout
x 100%
P
29
Procedure:
1. Record the rated values of the induction motor. Note the synchronous speed.
2. Couple the induction motor to the prony brake as shown in fig.1, adjust the
prony brake belt so that it is not very tight.
3. Connect the two wattmeters to read the total power.
4. Start the motor and perform a load to 5 Nm in steps of 0.5 Nm.
a
P1
M
A
3Φ ac
A
T
V1
ROTOR
B
b
C
A
c
INDUCTION MOTOR
Prony Brake
Fig.1: Connection of 3Φ Induction Motor
5. Prepare a table similar to table.1 on a separate sheet of paper. Record the
motor speed n (rpm) and load T(Nm) and the wattmeter readings P1 and P2
(watts).
Report
1. Calculate the total input power, the slip and the output power for each
Pout = 2 ( π / 60) Tn
Slip s = ( ns – n ) / ns
watts
ns = 1800 rpm ( syn. Speed ).
2.
3.
4.
5.
Plot torque vs speed and torque vs slip.
Calculate efficiency of the motor and enter it in table.1.
Plot efficiency vs torque.
Find maximum torque and slip conditions.
30
Table.1
Torque-T
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Speed
P1
P2
Ptotal
(P1+P2)
31
Slip
Pout (watts)
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
Electric Engineering Department
EE 306
Electric Energy Engineering-Experiment#10
DETERMINATION OF INDUCTION MOTOR PARAMETERS
Objective
The goal of this experiment is to determine the electrical parameters of a 3-ϕ
induction motor (primary and secondary resistance and reactance and the
magnetization branch values).
APPARATUS
1) 1 Three-Phase Induction Motor.
2) 1 Prony Brake.
3) 2 Digital Wattmeters.
4) 1 Three-Phase Variable AC Power Supply.
5) 1 DC Power Supply.
6) 1 DC Ammeter.
7) 1 DC Voltmeter.
8) 2 Three Phase Switches.
Introduction
Induction motor is an AC machine in which an alternating current is supplied to
the stator armature windings directly and to the rotor windings by induction.
Because it operates at balanced conditions, only a single phase is necessary. So,
the per-phase equivalent circuit of the induction motor in which the rotor
parameters are referred to the stator side is shown in Figure 1. It can be seen
from Figure 1 that the core loss represented by RC is neglected since its effect is
lumped with the rotational losses. The following equations can be derived:
V1 = E1 + (R1 + jX 1 )I 1 ................................................... (1)
⎛R
⎞
E1 = ⎜ 2 + jX 2 ⎟ I 2 ...................................................... (2)
⎝ s
⎠
To determine the parameters of the equivalent circuit of the three-phase
induction motor, it is subjected to three tests.
32
j X1
R1
I1
j X2
I2
+
+
j Xm
V1
R2 / s
E1
_
_
Figure 1: Per-phase equivalent circuit of a three-phase induction motor referred to the stator
Proceedure
A.
DC Test
Connect the circuit as shown in Figure 2 (while the motor is at standstill), apply
the dc voltage Vdc until the current Idc flowing in the induction motor is the rated
value. The stator resistance per phase can be calculated as R1 = Vdc / (2 Idc)..
I dc
A
V
Vdc
A
R1
R1
B
R1
C
Figure 2: DC test for the determination of the stator resistance
B.
Rated balanced voltage at rated frequency is applied to the stator, and the motor
is allowed to run on no-load. When the machine runs on no-load, the slip is close
to zero, and the circuit to the right of the shunt branch in Figure l is taken to be
an open circuit. Thus the equivalent circuit to the no-load test conditions is given
in Figure 3. Because of the relatively low value of rotor frequency, the rotor core
loss is practically negligible at no-load. From Figure 3, it follows that
Protational = Pnl − 3I nl2 R1 ................................................... (3)
P
Rnl = nl2 = R1 + lumped losses .................................... (4)
3 I nl
Vnl
Z nl =
= Rnl2 + X nl2 ............................................ (5)
3 I nl
X nl = Z nl2 − Rnl2 = X 1 + X m ......................................... (6)
33
No load power factor = cos ϕ 0 =
Inl
Pnl
3 Vnl I nl
..................... (7)
j X1
R1
+
j Xm
Vnl /sqrt(3)
_
Figure 3: Approximate equivalent circuit for no load test
Perform the following:
1.
2.
3.
4.
5.
Connect the circuit as shown in Figure 4. Apply the rated voltage.
Measure the rated voltage Vo = Vnl.
Measure the line current (Ia = Ib = Ic = Inl).
Measure the wattmeters powers W1 and W2, so Pnl = W1 + W2.
Calculate Rnl, Xnl, Znl, and φ0 from equations (4)−(7).
W1
a
Ia
b
A
V0
3 Phase Supply Rated
Voltage
Ib
c
B
C
Ic
W3
2
Figure 4: Schematic diagram for the no load test
C.
Blocked-Rotor Test
In this test, the rotor of the induction motor is blocked so that the slip is equal to
unity, and a reduced voltage value is applied to the machine stator terminals so
that the rated current flows through the stator windings. The iron losses are
assumed to be negligible in this test. Also, the shunt branch is neglected for this
test since the excitation current is small. The equivalent circuit corresponding to
the blocked rotor test condition is given in Figure 5. From Figure 5, it then
follows that
Rbl =
Pbl
= R1 + R2 .................................................... (8)
3 I bl2
34
Z bl =
Vbl
3 I bl
= Rbl2 + X bl2 ............................................ (9)
X bl = Z bl2 − Rbl2 = X 1 + X 2 .......................................... (10)
The following assumption can be taken:
X1 = X 2 =
1
X bl ......................................................... (11)
2
Finally, the magnetization reactance can be found:
X m = X nl − X 1 ............................................................ (12)
j X1
R1
Ibl
j X2
+
Vbl /sqrt(3)
R2
_
Figure 5: Approximate equivalent circuit for blocked rotor test
Perform the following:
1. Connect the circuit as in Figure 6. Keep the applied voltage to zero at
starting.
2. Increase the applied voltage until the rated current flows in the stator
winding.
3. Measure the applied voltage VS = Vbl.
4. Measure the line current (Ia = Ib = Ic = Ibl).
5. Measure the wattmeters powers W1 and W2, so Pbl = W1 + W2.
6. Apply equations 8−12 to calculate the parameters X1, X2 , Xm , R2 .
W1
a
Ia
b
A
V
VS
0
3 Phase
Supply
Ib
c
B
W32
Ic
Figure 6: Schematic diagram for the blocked-rotor test
35
C
Brake
Report
1. Record the ratings of the induction motor and determine the number of its
poles.
2. Find the parameters of the equivalent circuit of the three-phase induction
motor.
3. Draw the equivalent circuit of the induction motor and put the values of the
parameters that you found in the previous question along with their symbols.
4. Determine the no load power angle.
5. Determine the combined rotational losses of the motor.
36
KING FAHD UNIVERSITY OF PETROLEUM & MINERALS
Electric Engineering Department
EE 360 LAB – EXP # 1 & 4
1. Y CONNECTION
Pf
0.8
lag
480W+420W+120W
480W+420W
480W+120W
480W
480W+420W+420W
+all inductance (22.9)
mH
480W+420W+420W
+all capacitors (22.9)
mH
0.8
P1
(W)
P2
(W)
PT
(W)
VAB
(V)
VCB
(V)
IA
(A)
P.f
Cal.
2. DELTA (∆) CONNECTION
For the above combinations of loading in ∆ connection reduce the applied
voltage (60 V) to limit the line current (same as in Y connected load).
37
P.f
```
| 7,979 | 28,012 |
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| 3.59375 | 4 |
CC-MAIN-2018-26
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https://studdy.ai/shared-solution/97959c8d-799f-4ea3-b56d-f63aece3edeb
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# Math Snap
## The sum of two positive number is 20 . Find the numbers if the product of one and cube of the other is a maximum.
#### STEP 1
Assumptions 1. Let the two positive numbers be $x$ and $y$. 2. The sum of these numbers is given by $x + y = 20$. 3. We need to maximize the expression $x \cdot y^3$.
#### STEP 2
Express $y$ in terms of $x$ using the sum equation. $y = 20 - x$
#### STEP 3
Substitute $y$ in the expression we need to maximize. $f(x) = x \cdot (20 - x)^3$
#### STEP 4
Expand the expression $(20 - x)^3$ using the binomial theorem. $(20 - x)^3 = 20^3 - 3 \cdot 20^2 \cdot x + 3 \cdot 20 \cdot x^2 - x^3$
#### STEP 5
Substitute the expanded form back into $f(x)$. $f(x) = x \cdot (8000 - 1200x + 60x^2 - x^3)$
#### STEP 6
Expand $f(x)$ to form a polynomial. $f(x) = 8000x - 1200x^2 + 60x^3 - x^4$
#### STEP 7
To find the maximum value, take the derivative of $f(x)$ with respect to $x$ and set it to zero. $f'(x) = 8000 - 2400x + 180x^2 - 4x^3$
#### STEP 8
Solve the equation $f'(x) = 0$ for $x$. This is a cubic equation, and we are looking for real and positive solutions.
#### STEP 9
Use a numerical or graphical method to find the roots of the cubic equation, as solving it analytically might be complex. We can also use estimation and check values around the critical points.
#### STEP 10
Check the second derivative to confirm the nature of the critical points (maximum or minimum). $f''(x) = -2400 + 360x - 12x^2$
#### STEP 11
Evaluate $f''(x)$ at the critical points found in STEP_8 to determine if they correspond to a maximum.
#### STEP 12
Select the value of $x$ that maximizes $f(x)$ and calculate $y = 20 - x$.
##### SOLUTION
Verify that both $x$ and $y$ are positive and that they indeed maximize the product $x \cdot y^3$.
Was this helpful?
| 580 | 1,804 |
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| 4.6875 | 5 |
CC-MAIN-2024-22
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| 0.712512 |
https://metanumbers.com/1029215
| 1,643,274,855,000,000,000 |
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|
crawl-data/CC-MAIN-2022-05/segments/1642320305242.48/warc/CC-MAIN-20220127072916-20220127102916-00553.warc.gz
| 442,664,969 | 7,411 |
# 1029215 (number)
1,029,215 (one million twenty-nine thousand two hundred fifteen) is an odd seven-digits composite number following 1029214 and preceding 1029216. In scientific notation, it is written as 1.029215 × 106. The sum of its digits is 20. It has a total of 3 prime factors and 8 positive divisors. There are 748,480 positive integers (up to 1029215) that are relatively prime to 1029215.
## Basic properties
• Is Prime? No
• Number parity Odd
• Number length 7
• Sum of Digits 20
• Digital Root 2
## Name
Short name 1 million 29 thousand 215 one million twenty-nine thousand two hundred fifteen
## Notation
Scientific notation 1.029215 × 106 1.029215 × 106
## Prime Factorization of 1029215
Prime Factorization 5 × 11 × 18713
Composite number
Distinct Factors Total Factors Radical ω(n) 3 Total number of distinct prime factors Ω(n) 3 Total number of prime factors rad(n) 1029215 Product of the distinct prime numbers λ(n) -1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) -1 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0
The prime factorization of 1,029,215 is 5 × 11 × 18713. Since it has a total of 3 prime factors, 1,029,215 is a composite number.
## Divisors of 1029215
8 divisors
Even divisors 0 8 4 4
Total Divisors Sum of Divisors Aliquot Sum τ(n) 8 Total number of the positive divisors of n σ(n) 1.34741e+06 Sum of all the positive divisors of n s(n) 318193 Sum of the proper positive divisors of n A(n) 168426 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 1014.5 Returns the nth root of the product of n divisors H(n) 6.11078 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors
The number 1,029,215 can be divided by 8 positive divisors (out of which 0 are even, and 8 are odd). The sum of these divisors (counting 1,029,215) is 1,347,408, the average is 168,426.
## Other Arithmetic Functions (n = 1029215)
1 φ(n) n
Euler Totient Carmichael Lambda Prime Pi φ(n) 748480 Total number of positive integers not greater than n that are coprime to n λ(n) 93560 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 80452 Total number of primes less than or equal to n r2(n) 0 The number of ways n can be represented as the sum of 2 squares
There are 748,480 positive integers (less than 1,029,215) that are coprime with 1,029,215. And there are approximately 80,452 prime numbers less than or equal to 1,029,215.
## Divisibility of 1029215
m n mod m 2 3 4 5 6 7 8 9 1 2 3 0 5 5 7 2
The number 1,029,215 is divisible by 5.
## Classification of 1029215
• Arithmetic
• Deficient
• Polite
• Square Free
### Other numbers
• LucasCarmichael
• Sphenic
## Base conversion (1029215)
Base System Value
2 Binary 11111011010001011111
3 Ternary 1221021211002
4 Quaternary 3323101133
5 Quinary 230413330
6 Senary 34020515
8 Octal 3732137
10 Decimal 1029215
12 Duodecimal 41773b
20 Vigesimal 68d0f
36 Base36 m25b
## Basic calculations (n = 1029215)
### Multiplication
n×y
n×2 2058430 3087645 4116860 5146075
### Division
n÷y
n÷2 514608 343072 257304 205843
### Exponentiation
ny
n2 1059283516225 1090230484151513375 1122081567745999838250625 1154863180747699223525117009375
### Nth Root
y√n
2√n 1014.5 100.965 31.8513 15.9405
## 1029215 as geometric shapes
### Circle
Diameter 2.05843e+06 6.46675e+06 3.32784e+12
### Sphere
Volume 4.56675e+18 1.33113e+13 6.46675e+06
### Square
Length = n
Perimeter 4.11686e+06 1.05928e+12 1.45553e+06
### Cube
Length = n
Surface area 6.3557e+12 1.09023e+18 1.78265e+06
### Equilateral Triangle
Length = n
Perimeter 3.08764e+06 4.58683e+11 891326
### Triangular Pyramid
Length = n
Surface area 1.83473e+12 1.28485e+17 840351
| 1,346 | 3,968 |
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| 3.71875 | 4 |
CC-MAIN-2022-05
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latest
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