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https://modulomathy.com/2012/05/27/math-trivia-april4-solution/
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## #math trivia #April4 solution
Burt Kaliski Jr. (@modulomathy)
4/4/12 7:14 PM
#math trivia for #April4: How many times this year do month and day both divide into year (’12)? In what year does this happen most often?
Another non-day-number problem, similar to #April3.
Both day and month divide into year 36 times in ’12: whenever both are among the six values 1, 2, 3, 4, 6 or 12.
Here, a very highly divisible number can be useful in any month whose number divides it. Larger years may seem better, but are not necessarily so. For instance, 96 is divisible by seven numbers between 1 and 12, as well as by 16 and 24, so the number of month/day combinations that works is 7*9 = 63. Meanwhile, 90 is divisible by seven numbers between 1 and 12, as well as by 15, 18, and 30, so yields 7*10-1 = 69 combinations (the omission being February 30). And 60 is divisible by eight numbers between 1 and 12 as well as 15 and 30, and thus produces 8*10-1 = 79 combinations, which is the record.
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# BIGGER BRRRR! Does water expand when it freezes? Purple team 4 th grade Mrs. Johnson Northview Elementary School.
## Presentation on theme: "BIGGER BRRRR! Does water expand when it freezes? Purple team 4 th grade Mrs. Johnson Northview Elementary School."— Presentation transcript:
BIGGER BRRRR! Does water expand when it freezes? Purple team 4 th grade Mrs. Johnson Northview Elementary School
STATEMENT OF THE QUESTION Does water get larger, or expand, when it freezes?
PROJECT OVERVIEW We have been studying weathering in our science class. Weathering is when rock breaks down into smaller pieces. Our text says one of the causes of weathering is water that seeps into the cracks of rocks and freezes. When it freezes, it expands and breaks the rock. We will test this idea. We will measure the water in several containers, and then freeze it. We will then measure the water again.
RESEARCH The most important ideas from our research were: -Water can exist as a solid, liquid, or gas. -When water freezes, it changes from a liquid to a solid we call ‘ice’. -Freezing water can cause serious problems for people. For example, when water in water pipes freezes, it can cause the pipes to burst. -Most materials contract, or get smaller as they get colder, and expand, or get larger when they heat up.
VARIABLES CONTROLLED variables – these variables will be kept the same: - type of water - type of container - time in freezer - position in freezer - amount of water in container INDEPENDENT variable – this variable is the one we changed on purpose: - temperature
V VARIABLES, CONTINUED Dependent variable - level of water in graduated cylinder
PREDICTION OR HYPOTHESIS We predict that the water WILL expand because many of us have put plastic water bottles in the freezer. When we take them out the next day, sometimes the bottom of the bottle sticks out so the bottle can’t even stand up. Sometimes, the bottle actually breaks in the freezer. We think this is because the water expanded as it froze.
MATERIALS 6 graduated cylinders Water from the tap Black sharpie Freezer
PROCEDURE 1. Each table will fill a graduated cylinder with water up to the 20 ml mark. 2. Each table will label their cylinder with their table letter. 3. We will place all of the cylinders in the freezer in the teacher lounge. 4. We will leave the cylinders over night. 5. The next day we will remove the cylinders and measure the level of the water.
DATA/OBSERVATIONS Table A Table B Table C Table D Table E Table F Starting Height 20 ml Ending Height 22 ml21 ml22.5 ml22 ml21 ml23 ml Change+2+1+2.5+2+1+3 CHANGE IN HEIGHT OF WATER BEFORE AND AFTER FREEZING
CONCLUSION The level of water in all of our cylinders was higher after we froze the water. So we concluded that the water DID expand as it froze. This explains why water pipes can burst if the water in them freezes. We would like to continue our investigation by measuring the water level in our cylinders each day for 2 weeks after we remove them from the freezer. We hope to observe the process of evaporation in which the liquid water changes to gas.
WORKS CITED Black, Peter. Water Drops. New York: SUNY Press, 2012. Kids Konnect website: 2012. Online. 10 Sep. 2012. Available: http://www.kidskonnect.comhttp://www.kidskonnect.com Science for Kids website: 2012. Online. 10 Sep. 2012. Available: http://www.sciencekids.co.nzhttp://www.sciencekids.co.nz Wick, Walter. A Drop of Water. New York: Scholastic Press, 1997.
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# Leetcode ——Lowest Common Ancestor of a Binary Tree
### Question
Given a binary tree, find the lowest common ancestor (LCA) of two given nodes in the tree.
According to the definition of LCA on Wikipedia: “The lowest common ancestor is defined between two nodes p and q as the lowest node in T that has both p and q as descendants (where we allow a node to be a descendant of itself).”
Given the following binary tree: root = [3,5,1,6,2,0,8,null,null,7,4]
_______3______
/ \
___5__ ___1__
/ \ /
6 _2 0 8
/
7 4
Example 1:
Input: root = [3,5,1,6,2,0,8,null,null,7,4], p = 5, q = 1
Output: 3
Explanation: The LCA of of nodes 5 and 1 is 3.
Example 2:
Input: root = [3,5,1,6,2,0,8,null,null,7,4], p = 5, q = 4
Output: 5
Explanation: The LCA of nodes 5 and 4 is 5, since a node can be a descendant of itself
according to the LCA definition.
Note:
All of the nodes' values will be unique.
p and q are different and both values will exist in the binary tree.
### Code
/**
* Definition for a binary tree node.
* struct TreeNode {
* int val;
* TreeNode *left;
* TreeNode *right;
* TreeNode(int x) : val(x), left(NULL), right(NULL) {}
* };
*/
class Solution {
public:
TreeNode* lowestCommonAncestor(TreeNode* root, TreeNode* p, TreeNode* q) {
if (root == NULL || root == p || root == q) return root;
TreeNode* left = lowestCommonAncestor(root->left, p, q);
TreeNode* right = lowestCommonAncestor(root->right, p, q);
return left == NULL ? right : right == NULL ? left : root;
}
};
posted @ 2018-08-06 13:02 清水汪汪 阅读(153) 评论(0编辑 收藏 举报
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# Gravitational potential energy and work done
If an object is lifted, work is done against the force of gravity.
When work is done energy is transferred to the object and it gains gravitational potential energy.
If the object falls from that height, the same amount of work would have to be done by the force of gravity to bring it back to the Earth’s surface.
If an object at a certain height has 2000 J of gravitational potential energy, we can say that:
2000 J of work has been done in getting the object to that height from the ground
and
2000 J of work would have to be done to bring it back to the ground.
Gravitational potential energy = work done
Kinetic energy, gravitational potential energy and conservation of energy
If an object, such as a ball is lifted above the ground it has gravitational potential energy.
If the ball is then dropped from rest it will fall back to the ground. The gravitational potential energy is converted to .
Due to the Principle of Conservation of Energy we can say that:
Gravitational potential energy at the top = kinetic energy at the bottom
GPEtop = KEbottom
Question
A ball of mass 0.4 kg is lifted to a height of 2.5 m.
It is then dropped, from rest.
What is the speed of the ball as it hits the ground (g = 10 N/kg)
GPEtop = KEbottom
m = 0.4 kg
g = 10 N/kg
h = 2.5 m
(mgh)top = ($$\frac{1}{2}$$ mv2)bottom
0.4 kg x 10 N/kg x 2.5 m = $$\frac{1}{2}$$ x 0.4kg x v2
10 = 0.2 v2
v2 = $$\frac{10}{0.2}$$
v2 = 50
v = $$\sqrt{50}$$
v = 7.1 m/s
The ball returns to the ground with a speed of 7.1 m/s.
Question
1. A mass of 25 kg is dropped from the top of a tower 20 m high. What is the speed of the mass as it hits the ground?
2. A mass of 50 kg is then dropped from the same height. What is its speed as it hits the ground?
1. GPEtop = Kebottom
m = 25 kg
g = 10 N/kg
h = 20 m
(mgh)top = ($$\frac{1}{2}$$mv2)bottom
25 kg x 10 N/kg x 20 m = $$\frac {1}{2}$$ x 25 kg x v2
5000 = 12.5 v2
v2 = $$\frac{5000}{12.5}$$
v2 = 400
v = $$\sqrt{400}$$
v = 20 m/s
The 25 kg mass returns to the ground with a speed of 20 m/s.
2. GPEtop = Kebottom
m = 50 kg
g = 10 N/kg
h = 20 m
(mgh)top = ($$\frac{1}{2}$$mv2)bottom
50 kg x 10 N/kg x 20 m = $$\frac {1}{2}$$ x 50 kg x v2
10 000 = 25 v2
v2 = $$\frac{10,000}{25}$$
v2 = 400
v = $$\sqrt{400}$$
v = 20 m/s
The 50 kg mass returns to the ground with a speed of 20 m/s, which is exactly the same as the speed of the 25 kg mass.
If there was no air resistance or drag, the 25 kg mass and the 50 kg would fall at the same rate of 10 m/s2.
Dropped from the same height, they both hit the ground at the same speed and after the same period of time.
This is true of all objects regardless of their mass - in the absence of air resistance (friction) they fall at the same rate.
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# How do I calculate a percentage decrease?
## How do I calculate a percentage decrease?
Calculate Percentage Decrease: First, work out the difference (decrease) between the two numbers you are comparing. Next, divide the decrease by the original number and multiply the answer by 100. If the answer is a negative number, this is a percentage increase.
## What is the percentage decrease from 140 to 77?
Latest numbers decreased by percentage of value
140, percentage decreased by 77% (percent) of its value = 32.2 Jul 23 02:12 UTC (GMT)
2.04, percentage decreased by 37% (percent) of its value = 1.2852 Jul 23 02:12 UTC (GMT)
169.81, percentage decreased by 3% (percent) of its value = 164.7157 Jul 23 02:12 UTC (GMT)
What is the percent decrease from 77 to 65?
Percentage Calculator: What is the percentage increase/decrease from 77 to 65? = -15.58.
How do I calculate percentage over?
How do I calculate percentage increase over time?
1. Divide the larger number by the original number.
2. Subtract one from the result of the division.
3. Multiply this new number by 100.
4. Divide the percentage change by the period of time between the two numbers.
5. You now have the percentage increase over time.
### How do you decrease a percentage from a total?
How to Calculate Percentage Decrease
1. Subtract starting value minus final value.
2. Divide that amount by the absolute value of the starting value.
3. Multiply by 100 to get percent decrease.
4. If the percentage is negative, it means there was an increase and not an decrease.
### How do you increase 25% of 180?
1. (i) Increase 180 by 25%
2. New value = 180 + (180 × 25)/100.
3. = 180 + 45 = 225.
4. (ii) Decrease 140 by 18%
5. New value = 140 – (140 × 18)/100.
6. = 140 – (14 × 18)/10 = 140 – 126/5.
7. = 140 – 25.2 = 114.8.
What percentage increase is 65 to 77?
Percentage Calculator: What is the percentage increase/decrease from 65 to 77? = 18.46.
What is the percent of decrease from 6000 to 60?
Percentage Calculator: What is the percentage increase/decrease from 6000 to 60? = -99.
#### When to multiply by 100 to get percentage decrease?
Multiply by 100 to get percent decrease If the percentage is negative, it means there was an increase and not an decrease.
#### How to calculate percentage decrease in body weight?
Of course, it is easiest to use our online percentage decrease calculator, but if you want to do the math by hand, it is 100 – 150 / 160 * 100 = 100 – 0.9375 * 100 = 100 – 93.75 = 6.25 percent decrease in body weight.
What does it mean when the percentage decrease is negative?
If the percentage is negative, it means there was an increase and not an decrease. You can use the percentage decrease formula for any percent decrease calculation: You have a lamp with a 60-watt traditional light bulb. Your lamp uses 60 watts of electricity per hour.
How to calculate the percentage decrease in your income?
Suppose original value is 750 and new value is 590. Input this into the formula below. Perform the operation 750 – 590 = 160. Divide 160 by 750 to get 0.213.
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## Q. 2.11
Let the joint distribution of (X,Y) be such that the possible outcomes (u , v) are the points within the rectangle $(-1 \leq u \leq 2,- 1 \leq v \leq 1)$. Furthermore, let every one of these possible outcomes be “equally likely.” Note that we are not dealing with discrete random variables in this example, because the set of possible values is not discrete. As with a single random variable on a continuous set of possible values, the term equally likely denotes a constant value of the probability density function. Thus, we say that
$p_{XY}(u,v)=C$ for $-1 \leq u \leq 2,- 1 \leq v \leq 1)$ $p_{XY}(u,v)=0$ otherwise
in which C is some constant. Find the value of the constant C and the joint cumulative distribution function $F_{XY}(u,v)$ for all values of $(u,v)$.
## Verified Solution
We find the value of C by using the total probability property that (X,Y) must fall somewhere within the space of possible values and that the probability of being within a set is the double integral over that set, as in Eq. 2.21.
$P[(X,Y)\in A]=\int_{A}^{}{} \int_{}^{}{p_{XY}(u,v)du dv}$
Thus, we obtain C = 1/6, because the rectangle of possible values has an area of 6 and the joint probability density function has the constant value C everywhere in the rectangle:
$1=P(-1 \leq X \leq 2,-1 \leq Y \leq 1)= \int_{-1}^{1}{} \int_{-1}^{2}{p_{XY}(u,v)du dv}=6C$
We may now calculate the joint cumulative distribution of these two random variables by integrating as in Eq. 2.20,
$F_{\vec{X}}(\vec{u})=\int_{-\infty}^{u_{n}}{…} \int_{-\infty}^{u_{1}}{p_{\vec{X}}(\vec{w})dw_{1}…dw_{n}}$
with the result that
$F_{XY}(u,v)=C(u+1)(v+1)=(u+1)(v+1)/6$ for $-1 \leq u \leq 2,- 1 \leq v \leq 1$ $F_{XY}(u,v)=3C(v+1)=(v+1)/2$ for $u \gt 2,- 1 \leq v \leq 1$ $F_{XY}(u,v)=2C(u+1)=(u+1)/3$ for $-1 \leq u \leq 2, v \gt 1$ $F_{XY}(u,v)=6C=1$ for $u \gt 2, v \gt 1$ $F_{XY}(u,v)=0$ otherwise
One can verify that this cumulative distribution function is continuous. Thus, the random variables X and Y are said to have a continuous joint distribution. Clearly, for this particular problem the description given by the density function is much simpler than that given by the cumulative distribution function.
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## Calculus: Early Transcendentals 8th Edition
Published by Cengage Learning
# APPENDIX E - Sigma Notation - E Exercises: 24
#### Answer
$1$
#### Work Step by Step
We expand and simplify the sum: $\displaystyle \sum_{k=0}^{8}\cos k\pi=\cos 0\pi+\cos1\pi+\cos 2\pi+\cos 3\pi+\cos 4\pi+\cos 5\pi+\cos 6\pi+\cos 7\pi+\cos 8\pi$ $=1-1+1-1+1-1+1-1+1$ $=1$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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# Hex grid cells in range
May 9, 2018
Getting the cells in a range of N is also an important aspect that worth talking about. There are 2 current uses for this feature in Hexarategy:
• get the cells that a weapon of range N can target
• get the cells the ship can navigate to
The algorithm implementations will depend on the underlying data structure you choose to represent you grid. In this post I will go over algorithms that can be used when using a graph to represent the grid. Let’s start with the weapon range, as the movement range is a bit more complicated since it has some different constrains.
The range of a weapon is defined as all the cells in all directions from the home cell that are < N units away. Think like the inside area of a circle.
If the yellow cell is the home cell, then the purple cells are all the cells that are 1 units away, and the orange cells + purple cells are all the cells that are 2 units away. So given the yellow colored home cell, how can we get the cells that are N units away?
This is a classical graph walk with a range constraint. Instead of visiting all nodes, or quitting when a node is found, we need to quit when we exhaust our range. So we need to keep track of the range, that’s for sure. We also need to consider when we decrease this range, because for all purple cells the range from the home cell is 1, and we cannot decrease the range each time we visit a purple cell. What we need is to keep track of how many purple cells there are and decrease the range after we have consumed them all. We will need track of this by using 2 extra counters: nodesInNext and nodesInCurrent . nodesInNext will hold the count of the neighbors in the next level and the current will hold how much of the current level we have consumed. There are recursive graph traversing algorithms and there are ones using stacks and queues. We’ll opt for a non-recursive algorithm here
``````... get the home cell and add it to the queue
goneRange = N //how far is our range?
nodesInCurrent = 1 /*home cell*/, nodesInNext = 0;
while(queue.Count > 0 && goneRange > 0)
{
nodesInCurrent--;
currentCell = queue.Dequeue();
List<Cell> neighborList = new List<Cell>();
foreach (Cell item in currentCell.neighbors)
{
}
neighborList.Remove(homeCell);
IEnumerable<HexCell> nlist = neighborList.Except(result);
nodesInNext += nlist.Count();
foreach (Cell neighbor in nlist)
{
queue.Enqueue(neighbor);
}
if (nodesInCurrent == 0)
{
goneRange--;
nodesInCurrent = nodesInNext;
nodesInNext = 0;
}
}
``````
the result list will contain all the cells that are in range and we can use that. Note that this doesn’t check for friendly fire.
The movement range calculations are bit more interesting as there is the condition that a rotation move also costs 1 range. Take a look at this diagram
Starting at the orange cell facing NE the yellow cell is in range 1. The cell NE of the yellow cell is in range 2 as it is in the same direction. The cells to the NW and E of the orange cell are in range 2 because 1 range point would be used to rotate the ship from NE to NW and NE to E. Let’s solve this algorithm with a recursive approach. Here is the outline of what we will try to implement
• if no more range points left, return
• if the result set doesn’t contain the current cell, add it
• for the next 3 directions, if there is a neighbor cell in that direction recurse with range – 1 in that direction otherwise decrease range points.
• for the previous 2 directions, if there is a neighbor cell in that direction recurse with range – 1 in that direction otherwise decrease range points.
So if we are facing NE like the diagram, and we have 2 range points, we will go to the yellow neighbor. We still have have range point so we’ll go NE. Now we don’t have any range points so add this cell to the results. Backtrack to the yellow cell and decrease the range points. The next direction is E, but we don’t have any range points. This will hold for all the remaining directions on the this cell, so we backtrack to the orange cell. The next direction is E, and we consumed a range point by turning in this direction so we have 1 point left. There is a cell to the E so go to that cell and add it to the list… you get the idea.
``````public void CellRangeMovement(Cell currentCell, int rangeLeft, List<Cell> result, Direction direction)
{
if (rangeLeft < 0) return;
if (!result.Contains(currentCell))
{
}
Direction nextDirection = direction;
int rangeReg = rangeLeft;
for (int i = 0; i < 3; i++)
{
Cell currentNeighbor = currentCell.neighbors[nextDirection];
if (currentNeighbor != null)
{
CellRangeMovement(currentNeighbor, rangeReg-1, result, nextDirection);
}
nextDirection = nextDirection.Next();
rangeReg--;
}
rangeReg = rangeLeft;
nextDirection = direction;
for (int i = 1; i < 3; i++)
{
Cell currentNeighbor = currentCell.neighbors[nextDirection];
if (currentNeighbor != null)
{
CellRangeMovement(currentNeighbor, rangeReg - 1, result, nextDirection);
}
nextDirection = nextDirection.Prev();
rangeReg--;
}
}
``````
One interesting point here is that, you cannot rotate 6 six times clockwise to get all the directions, as that would mean consuming extra ranges, e.g if you are facing NE and want to face NW you would not go E,SE,SW,W,NW. You would just go NE, NW. That’s why you have to check the directions in 2 separate for loops.
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# Math
posted by .
Given f(x)= |x-1|/x^2-25 please determine the domain. Thank you!
A. All real numbers except -5 and 5
B. All real numbers except -5,0 and 5
C. All real numbers except 0
D. All real numbers except 0 and 5
• Math -
the only thing you have to worry about in your function is division by zero
When does that happen?
When x^2 - 25 = 0
or when x^2 = 25
or when x = ± 5
so your domain is any real number, except x = ±5
which would be a)
• Math -
Thank you for the explanation! My teacher really never explained it like you did. Do you teach math?
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# A fishery stocks a pond with 1000 young trout. The number of trout t years later is given by Q(t) = 1000e(−0.5t). Graph the number of trout against time. Do this on extra paper.How many trout are left after 1 year ____________________________?Find Q(3) and interpret your answer in terms of trout _____________________?At what time are there 100 trout left _________________________________? a.A. Approximately 606 Trout; B. Approximately 223 and means that after 3 years the pond will have about 223 trout left. C. Approximately 4.6 years b.A. Approximately 606 Trout; B. Approximately 223 and means that after 3 years the pond will have about 223 trout left. C. Approximately 1.6 years c.A. Approximately 606 Trout; B. Approximately 2231 and means that after 3 years the pond will have about 223 trout left. C. Approximately 4.6 years d.A. Approximately 900 Trout; B. Approximately 223 and means that after 3 years the pond will have about 223 trout left. C. Approximately 1.6 years e.None of these
Question
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A fishery stocks a pond with 1000 young trout. The number of trout t years later is given by Q(t) = 1000e(−0.5t). Graph the number of trout against time. Do this on extra paper.
1. How many trout are left after 1 year ____________________________?
2. Find Q(3) and interpret your answer in terms of trout _____________________?
3. At what time are there 100 trout left _________________________________?
a. A. Approximately 606 Trout; B. Approximately 223 and means that after 3 years the pond will have about 223 trout left. C. Approximately 4.6 years b. A. Approximately 606 Trout; B. Approximately 223 and means that after 3 years the pond will have about 223 trout left. C. Approximately 1.6 years c. A. Approximately 606 Trout; B. Approximately 2231 and means that after 3 years the pond will have about 223 trout left. C. Approximately 4.6 years d. A. Approximately 900 Trout; B. Approximately 223 and means that after 3 years the pond will have about 223 trout left. C. Approximately 1.6 years e. None of these
check_circle
Step 1
(a)
Find the number of trout are left after 1 year.
Step 2
(b)
Find the number of trout are left after 3 years.
Step 3
(c)
Find the time when there a...
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https://www.sarthaks.com/202840/cost-cultivating-square-field-rate-hectare-3240-what-cost-putting-fence-around-paise-metre
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# The cost of cultivating a square field at the rate Rs 360 per hectare is Rs 3240. What is the cost of putting a fence around it at 75 paise per metre?
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edited
The cost of cultivating a square field at the rate Rs 360 per hectare is Rs 3240. What is the cost of putting a fence around it at 75 paise per metre?
commented by (10.4k points)
Dont forget to explain.
answered by (9.2k points)
selected
Rs 360 is the cost of cultivation of 1 hectare
So Rs 3260 will be cost for 3240/360=9hectare= 90000sqmeter
Since 1 hectare = 10000sqmeter
So area of square land =90000sqmeter
So length of each side = ✓90000=300m
Its perimeter =4*300=1200m
So the cost of fencing @Rs0.75 will be
=RS.0.75*1200=Rs900.
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# How do you solve 5x+6-3x=2x+5-1x?
Apr 25, 2018
x = -1
#### Explanation:
First, reorder and simplify the equation.
5x - 3x + 6 = 2x - 1x + 5
2x + 6 = x + 5. Subtract 5 from both sides.
2x + 1 = x. Subtract 2x from both sides.
1 = -x Multiply both sides by -1
x = -1.
Check:
$\left(5 \cdot - 1\right) + 6 - \left(3 \cdot - 1\right) = 4$
$\left(2 \cdot - 1\right) + 5 - \left(1 \cdot - 1\right) = 4$
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https://www.key2physics.org/ramps-inclines
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Ramps and Inclines
Some physics problems include defining forces and motion of objects in a ramp or incline. To work on these kind of problems, the following must be done.
1. Represent all the forces acting on the object on a ramp using the Free-Body Diagram. Use a box or a dot to represent an object and arrows representing the forces acting on the object with the head pointing at the direction of the force.
2. Apply Newton’s laws of motion and the concept of forces to solve problems of objects on the ramp.
3. Label all necessary components in the FBD.
Example 1.
An object is at rest in a ramp inclined at $$30^\circ$$ from the ground. What are the forces that act on the object?
FBD:
Example 2.
A $$150-kg$$ box is pushed towards the truck. If the ramp is inclined $$60^\circ$$ from the ground, how much force is applied to make the box moving at $$1.5\;m/s^2$$?
Given:
$$m= 150\;kg\\ \text{angle}=60^\circ\\ a=1.5\;m/s^2$$
Solution:
To solve for the force exerted on the box, we will use the Newton’s second law of motion $$F=ma$$.
$$F=(150\;kg)(1.5\;m/s^2)=225\;N$$
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# Finding Major and Minor Strain from Mohr Circle of Stress
• mjocalla
In summary, the conversation is about using the Mohr circle of stress to find the major and minor strains. One person asks if they can use Young's Modulus equation and value to find the strains, while the other person suggests using the principal stresses to determine the corresponding principal strains. They also mention that the only difference between Mohr's Circle for plane stress and plane strain is that the Y axis for strain is half of the shear strain.
#### mjocalla
Hey everyone.
Quick question about the Mohr circle.
If I have got the major and minor stress from a Mohr Circle of stress how do you go about finding the major and minor strains?
Can you just use Young's Modulus equation and value for Young's Modulus or will I need to construct Mohr Circle of Strain?
Thanks for any help.
mjocalla said:
Can you just use Young's Modulus equation and value for Young's Modulus or will I need to construct Mohr Circle of Strain?
I don't see why you would construct another Mohr circle. If you have Young's modulus given, and if you know the stresses, you can find the strain.
The question I was trying to ask was would the major/minor strain occur when the stress corresponds to the major/minor stress and vice versa?
Yes. The principal strains will lie in the same direction as the principal stresses. The only real difference between the Mohr's Circle for plane stress and plane strain is that for the circle in strain, the Y axis (shear strain) is equal to half of the shear strain.
## 1. What is the purpose of using the Mohr circle of stress?
The Mohr circle of stress is a graphical representation used to determine the state of stress at a specific point in a material. It helps in understanding the stress distribution and identifying the principal stresses and their corresponding directions.
## 2. What is major and minor strain?
Major and minor strain refer to the two principal strains that occur in a material under stress. Major strain is the largest strain in a material, while minor strain is the smallest. They are perpendicular to each other and represent the maximum and minimum elongation or compression of a material in different directions.
## 3. How do you find major and minor strain from the Mohr circle of stress?
To find major and minor strain from the Mohr circle of stress, you need to plot the given stress values on the circle and draw a line connecting them. The point where this line intersects the circle represents the center of the circle and the major and minor strains can be read from the horizontal and vertical axes, respectively.
## 4. What is the relationship between stress and strain in the Mohr circle?
In the Mohr circle, stress is represented by the distance from the center of the circle to the plotted point and strain is represented by the angle between the stress line and the horizontal axis. This relationship helps in understanding the stress-strain behavior of a material and determining its strength and stiffness.
## 5. How can the Mohr circle of stress be used in real-world applications?
The Mohr circle of stress is widely used in engineering and geology to analyze the stress and strain distribution in different types of structures and materials. It is also used in designing and testing materials for various applications, such as in construction, manufacturing, and geological studies.
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$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
# 4.3: Absolute Value Equations
$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$
$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$
In the previous section, we defined
$|x|=\left\{\begin{array}{ll}{-x,} & {\text { if } x<0} \\ {x,} & {\text { if } x \geq 0}\end{array}\right.$
and we saw that the graph of the absolute value function defined by f(x) = |x| has the “V-shape” shown in Figure $$\PageIndex{1}$$.
It is important to note that the equation of the left-hand branch of the “V” is y = −x. Typical points on this branch are (−1, 1), (−2, 2), (−3, 3), etc. It is equally important to note that the right-hand branch of the “V” has equation y = x. Typical points on this branch are (1, 1), (2, 2), (3, 3), etc.
## Solving |x| = a
We will now discuss the solutions of the equation
$|x|=a$
There are three distinct cases to discuss, each of which depends upon the value and sign of the number a.
• Case I: a < 0
If a < 0, then the graph of y = a is a horizontal line that lies strictly below the x-axis, as shown in Figure $$\PageIndex{2}$$(a). In this case, the equation |x| = a has no solutions because the graphs of y = a and y = |x| do not intersect.
• Case II: a = 0
If a = 0, then the graph of y = 0 is a horizontal line that coincides with the x-axis, as shown in Figure $$\PageIndex{2}$$(b). In this case, the equation |x| = 0 has the single solution x = 0, because the horizontal line y = 0 intersects the graph of y = |x| at exactly one point, at x = 0.
• Case III: a > 0
If a > 0, then the graph of y = a is a horizontal line that lies strictly above the x-axis, as shown in Figure $$\PageIndex{2}$$(c). In this case, the equation |x| = a has two solutions, because the graphs of y = a and y = |x| have two points of intersection.
Recall that the left-hand branch of y = |x| has equation y = −x, and points on this branch have the form (−1, 1), (−2, 2), etc. Because the point where the graph of y = a intersects the left-hand branch of y = |x| has y-coordinate y = a, the x-coordinate of this point of intersection is x = −a. This is one solution of |x| = a.
Recall that the right-hand branch of y = |x| has equation y = x, and points on this branch have the form (1, 1), (2, 2), etc. Because the point where the graph of y = a intersects the right-hand branch of y = |x| has y-coordinate y = a, the x-coordinate of this point of intersection is x = a. This is the second solution of |x| = a.
This discussion leads to the following key result.
Property 2
The solution of |x| = a depends upon the value and sign of a.
• Case I: a < 0
The equation |x| = a has no solutions.
• Case II: a = 0
The equation |x| = 0 has one solution, x = 0.
• Case III: a > 0
The equation |x| = a has two solutions, x = −a or x = a.
Let’s look at some examples.
Example $$\PageIndex{1}$$
Solve |x| = −3 for x.
Solution
The graph of the left-hand side of |x| = −3 is the “V” of Figure $$\PageIndex{2}$$(a). The graph of the right-hand side of |x| = −3 is a horizontal line three units below the x-axis. This has the form of the sketch in Figure $$\PageIndex{2}$$(a). The graphs do not intersect. Therefore, the equation |x| = −3 has no solutions.
An alternate approach is to consider the fact that the absolute value of x can never equal −3. The absolute value of a number is always nonnegative (either zero or positive). Hence, the equation |x| = −3 has no solutions.
Example $$\PageIndex{2}$$
Solve |x| = 0 for x
Solution
This is the case shown in Figure $$\PageIndex{2}$$(b). The graph of the left-hand side of |x| = 0 intersects the graph of the right-hand side of |x| = 0 at x = 0. Thus, the only solution of |x| = 0 is x = 0.
Thinking about this algebraically instead of graphically, we know that 0 = 0, but there is no other number with an absolute value of zero. So, intuitively, the only solution of |x| = 0 is x = 0.
Example $$\PageIndex{3}$$
Solve |x| = 4 for x.
Solution
The graph of the left-hand side of |x| = 4 is the “V” of Figure $$\PageIndex{2}$$(c). The graph of the right-hand side is a horizontal line 4 units above the x-axis. This has the form of the sketch in Figure $$\PageIndex{2}$$(c). The graphs intersect at (−4, 4) and (4, 4). Therefore, the solutions of |x| = 4 are x = −4 or x = 4.
Alternatively, | − 4| = 4 and |4| = 4, but no other real numbers have absolute value equal to 4. Hence, the only solutions of |x| = 4 are x = −4 or x = 4.
Example $$\PageIndex{4}$$
Solve the equation |3 − 2x| = −8 for x.
Solution
If the equation were |x| = −8, we would not hesitate. The equation |x| = −8 has no solutions. However, the reasoning applied to the simple case |x| = −8 works equally well with the equation |3 − 2x| = −8. The left-hand side of this equation must be nonnegative, so its graph must lie above or on the x-axis. The right-hand side of |3−2x| = −8 is a horizontal line 8 units below the x-axis. The graphs cannot intersect, so there is no solution.
We can verify this argument with the graphing calculator. Load the left and righthand sides of |3 − 2x| = −8 into Y1 and Y2, respectively, as shown in Figure $$\PageIndex{3}$$(a). Push the MATH button on your calculator, then right-arrow to the NUM menu, as shown in Figure $$\PageIndex{3}$$(b). Use 1:abs( to enter the absolute value shown in Y1 in Figure $$\PageIndex{3}$$(a). From the ZOOM menu, select 6:ZStandard to produce the image shown in Figure $$\PageIndex{3}$$(c).
Note, that as predicted above, the graph of y = |3 − 2x| lies on or above the xaxis and the graph of y = −8 lies strictly below the x-axis. Hence, the graphs cannot intersect and the equation |3 − 2x| = −8 has no solutions.
Alternatively, we can provide a completely intuitive solution of |3 − 2x| = −8 by arguing that the left-hand side of this equation is nonnegative, but the right-hand side is negative. This is an impossible situation. Hence, the equation has no solutions.
Example $$\PageIndex{5}$$
Solve the equation |3 − 2x| = 0 for x.
Solution
We have argued that the only solution of |x| = 0 is x = 0. Similar reasoning points out that |3 − 2x| = 0 only when 3 − 2x = 0. We solve this equation independently.
\begin{aligned} 3-2 x &=0 \\-2 x &=-3 \\ x &=\frac{3}{2} \end{aligned}
Thus, the only solution of |3 − 2x| = 0 is x = 3/2.
It is worth pointing out that the “tip” or “vertex” of the “V” in Figure $$\PageIndex{3}$$(c) is located at x = 3/2. This is the only location where the graphs of y = |3 − 2x| and y = 0 intersect.
Example $$\PageIndex{6}$$
Solve the equation |3 − 2x| = 6 for x.
Solution
In this example, the graph of y = 6 is a horizontal line that lies 6 units above the x-axis, and the graph of y = |3 − 2x| intersects the graph of y = 6 in two locations. You can use the intersect utility to find the points of intersection of the graphs, as we have in Figure $$\PageIndex{4}$$(b) and (c).
We need a way of summarizing this graphing calculator approach on our homework paper. First, draw a reasonable facsimile of your calculator’s viewing window on your homework paper. Use a ruler to draw all lines. Complete the following checklist.
• Label each axis, in this case with x and y.
• Scale each axis. To do this, press the WINDOW button on your calculator, then report the values of xmin, xmax, ymin, and ymax on the appropriate axis.
• Label each graph with its equation.
• Drop dashed vertical lines from the points of intersection to the x-axis. Shade and label these solutions of the equation on the x-axis.
Following the guidelines in the above checklist, we obtain the image in Figure $$\PageIndex{5}$$.
Algebraic Approach. One can also use an algebraic technique to find the two solutions of |3 − 2x| = 6. Much as |x| = 6 has solutions x = −6 or x = 6, the equation
$|3-2 x|=6$
is possible only if the expression inside the absolute values is either equal to −6 or 6. Therefore, write
$3-2 x=-6 \qquad \text { or } \qquad 3-2 x=6$
and solve these equations independently
$\begin{array}{rlrrrl}{3-2 x}&{=}&{-6} & {\text { or }} & {3-2 x}&{=}&{6} \\ {-2 x}&{=}&{-9} && {-2 x}&{=}&{3} \\ {x}&{=}&{\frac{9}{2}} && {x}&{=}&{-\frac{3}{2}}\end{array}$
Because −3/2 = −1.5 and 9/2 = 4.5, these exact solutions agree exactly with the graphical solutions in Figure $$\PageIndex{4}$$(b) and (c).
Let’s summarize the technique involved in solving this important case.
Note
Solving |expression| = a, when a > 0. To solve the equation
$| \text { expression } |=a, \quad \text { when } a>0$
set
$\text { expression }=-a \qquad \text { or } \qquad \text { expression }=a$
then solve each of these equations independently.
For example:
• To solve |2x + 7| = 5, set $2x + 7 = −5 \qquad or \qquad 2x + 7 = 5$, then solve each of these equations independently.
• To solve |3 − 5x| = 9, set $3 − 5x = −9 \qquad or \qquad 3 − 5x = 9$, then solve each of these equations independently.
• Note that this technique should not be applied to the equation |2x + 11| = −10, because the right-hand side of the equation is not a positive number. Indeed, in this case, no values of x will make the left-hand side of this equation equal to −10, so the equation has no solutions.
Sometimes we have to do a little algebra before removing the absolute value bars.
Example $$\PageIndex{7}$$
Solve the equation $|x+2|+3=8$ for x.
Solution
First, subtract 3 from both sides of the equation. \begin{aligned}|x+2|+3 &=8 \\|x+2|+3-3 &=8-3 \end{aligned}
This simplifies to $|x+2|=5$
Now, either $x+2=-5 \qquad \text { or } \qquad x+2=5$
each of which can be solved separately.
$\begin{array}{rrlrrl}{x+2} & {=} & {-5} & {\text { or }} & {x+2} & {=} & {5} \\ {x+2-2} & {=} & {-5-2} && {x+2-2} & {=} & {5-2} \\ {x} & {=} & {-7} && {x} & {=} & {3}\end{array}$
Example $$\PageIndex{8}$$
Solve the equation $3|x-5|=6$ for x.
Solution
First, divide both sides of the equation by 3
\begin{aligned} 3|x-5| &=6 \\ \frac{3|x-5|}{3} &=\frac{6}{3} \end{aligned}
This simplifies to $|x-5|=2$
Now, either $x-5=-2 \qquad \text { or } \qquad x-5=2$
each of which can be solved separately.
$\begin{array}{rllrrl}{x-5} & {=} & {-2} & {\text { or }} & {x-5} & {=} & {2} \\ {x-5+5} & {=} & {-2+5} && {x-5+5} & {=} & {2+5} \\ {x} & {=} & {3} && {x} & {=} & {7}\end{array}$
## Properties of Absolute Value
An example will motivate the need for some discussion of the properties of absolute value.
Example $$\PageIndex{9}$$
Solve the equation $\left|\frac{x}{2}-\frac{1}{3}\right|=\frac{1}{4}$ for x.
Solution
It is tempting to multiply both sides of this equation by a common denominator as follows.
$\begin{array}{l}{\left|\dfrac{x}{2}-\dfrac{1}{3}\right|=\dfrac{1}{4}} \\ {12\left|\dfrac{x}{2}-\dfrac{1}{3}\right|=12\left(\dfrac{1}{4}\right)}\end{array}$
If it is permissible to move the 12 inside the absolute values, then we could proceed as follows.
\begin{aligned}\left|12\left(\frac{x}{2}-\frac{1}{3}\right)\right| &=3 \\|6 x-4| &=3 \end{aligned}
Assuming for the moment that this last move is allowable, either
$6 x-4=-3 \qquad \text { or } \qquad 6 x-4=3$
Each of these can be solved separately, first by adding 4 to both sides of the equations, then dividing by 6.
$\begin{array}{rllrrl}{6 x-4} & {=} & {-3} & {\text { or }} & {6 x-4} & {=} & {3} \\ {6 x} & {=} & {1} & &{6 x} & {=} & {7} \\ {x} & {=} & {1 / 6} && {x} & {=} & {7 / 6}\end{array}$
As we’ve used a somewhat questionable move in obtaining these solutions, it would be wise to check our results. First, substitute x = 1/6 into the original equation.
\begin{aligned}\left|\frac{x}{2}-\frac{1}{3}\right| &=\frac{1}{4} \\\left|\frac{1 / 6}{2}-\frac{1}{3}\right| &=\frac{1}{4} \\\left|\frac{1}{12}-\frac{1}{3}\right| &=\frac{1}{4} \end{aligned}
Write equivalent fractions with a common denominator and subtract.
\begin{aligned}\left|\frac{1}{12}-\frac{4}{12}\right| &=\frac{1}{4} \\\left|-\frac{3}{12}\right| &=\frac{1}{4} \\\left|-\frac{1}{4}\right| &=\frac{1}{4} \end{aligned}
Clearly, x = 1/6 checks. We’ll leave the check of the second solution to our readers.
Well, we’ve checked our solutions and they are correct, so it must be the case that
$12\left|\frac{x}{2}-\frac{1}{3}\right|=\left|12\left(\frac{x}{2}-\frac{1}{3}\right)\right|$
But why? After all, absolute value bars, though they do act as grouping symbols, have a bit more restrictive meaning than ordinary grouping symbols such as parentheses, brackets, and braces.
We state the first property of absolute values.
Property
If a and b are any real numbers, then
$|a b|=|a||b|$
We can demonstrate the validity of this property by simply checking cases.
• If a and b are both positive real numbers, then so is ab and $$|a||b|=a b$$. On the other hand, $$|a||b|=a b$$. Thus, $$|ab| = |a||b|$$.
• If a and b are both negative real numbers, then ab is positive and $$|ab| = ab$$. On the other hand, $$|a||b| = (−a)(−b) = ab$$. Thus, $$|ab| = |a||b|$$.
We will leave the proof of the remaining two cases as exercises. We can use $$|a||b| = |ab|$$ to demonstrate that
$12\left|\frac{x}{2}-\frac{1}{3}\right|=|12|\left|\frac{x}{2}-\frac{1}{3}\right|=\left|12\left(\frac{x}{2}-\frac{1}{3}\right)\right|$
This validates the method of attack we used to solve equation (12) in Example $$\PageIndex{9}$$.
Warning 14
On the other hand, it is not permissible to multiply by a negative number and simply slide the negative number inside the absolute value bars. For example,
$-2|x-3|=|-2(x-3)|$
is clearly an error (well, it does work for x = 3). For any x except 3, the lefthand side of this result is a negative number, but the right-hand side is a positive number. They are clearly not equal.
In similar fashion, one can demonstrate a second useful property involving absolute value.
Definition
If a and b are any real numbers, then $\left|\frac{a}{b}\right|=\frac{|a|}{|b|}$ provided, of course, that $$b \neq 0$$.
Again, this can be proved by checking four cases. For example, if a is a positive real number and b is a negative real number, then a/b is negative and $$|a/b| = −a/b$$. On the other hand, $$|a|/|b| = a/(−b) = −a/b$$.
We leave the proof of the remaining three cases as exercises.
This property is useful in certain situations. For example, should you desire to divide $$|2x − 4|$$ by 2, you would proceed as follows.
$\frac{|2 x-4|}{2}=\frac{|2 x-4|}{|2|}=\left|\frac{2 x-4}{2}\right|=|x-2|$
This technique is useful in several situations. For example, should you want to solve the equation $$|2x − 4| = 6$$, you could divide both sides by 2 and apply the quotient property of absolute values.
## Distance Revisited
Recall that for any real number x, the absolute value of x is defined as the distance between the real number x and the origin on the real line. In this section, we will push this distance concept a bit further.
Suppose that you have two real numbers on the real line. For example, in the figure that follows, we’ve located 3 and −2 on the real line.
You can determine the distance between the two points by subtracting the number on the left from the number on the right. That is, the distance between the two points is d = 3 − (−2) = 5 units. If you subtract in the other direction, you get the negative of the distance, as in −2 − 3 = −5 units. Of course, distance is a nonnegative quantity, so this negative result cannot represent the distance between the two points. Consequently, to find the distance between two points on the real line, you must always subtract the number on the left from the number on the right.
However, if you take the absolute value of the difference, you’ll get the correct result regardless of the direction of subtraction.
$d=|3-(-2)|=|5|=5 \quad \text { and } \quad d=|-2-3|=|-5|=5$
This discussion leads to the following key idea.
Property 16.
Suppose that a and b are two numbers on the real line
You can determine the distance d between a and b on the real line by taking the absolute value of their difference. That is,
$d=|a-b|$
Of course, you could subtract in the other direction, obtaining $$d = |b − a|$$. This is also correct.
Now that this geometry of distance has been introduced, it is useful to pronounce the symbolism |a−b| as “the distance between a and b” instead of saying “the absolute value of a minus b.”
Example $$\PageIndex{10}$$
Solve the equation $|x − 3| = 8$ for x.
Solution
Here’s the ideal situation to apply our new concept of distance. Instead of saying “the absolute value of x minus 3 is 8,” we pronounce the equation $$|x − 3| = 8$$ as “the distance between x and 3 is 8.”
Draw a number line and locate the number 3 on the line.
Recall that the “distance between x and 3 is 8.” Having said this, mark two points on the real line that are 8 units away from 3.
Thus, the solutions of |x − 3| = 8 are x = −5 or x = 11
Example $$\PageIndex{11}$$
Solve the equation $|x + 5| = 2$ for x.
Solution
Rewrite the equation as a difference. $|x − (−5)| = 2$ This is pronounced “the distance between x and −5 is 2.” Locate two points on the number line that are 2 units away from −5.
Hence, the solutions of $$|x + 5| = 2$$ are x = −7 or x = −3.
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1. ## Venn Diagram Problem
Hi guys, please give me some clues into solving this question from a workbook.
Question: There are 50 units in an apartment block. 40 units subscribe to the Straits Times, 20 units subscribe to the Business Times, m units subscribe to both Straits Times and Business Times, and n units subscribe to neither Straits Times nor Business Times.
By drawing Venn diagrams, find
a) the smallest possible value of n,
b) the largest possible value of n,
c) the largest possible value of m,
d) the smallest possible value of m.
Thanks a lot!
2. ## Re: Venn Diagram Problem
A picture really does speak a thousand words when it comes to Venn diagrams. What have you tried?
Surely the smallest possible value of n is 0? In other words everyone subscribes to one or the other.
3. ## Re: Venn Diagram Problem
Originally Posted by e^(i*pi)
A picture really does speak a thousand words when it comes to Venn diagrams. What have you tried?
Surely the smallest possible value of n is 0? In other words everyone subscribes to one or the other.
Well, I would had been able to solve if there were only one unknown, but now there are two unknowns and i am totally clueless =(
4. ## Re: Venn Diagram Problem
Originally Posted by FailInMaths
Well, I would had been able to solve if there were only one unknown, but now there are two unknowns and i am totally clueless =(
Surely you can see that $\displaystyle 60-m+n=50$.
If both $\displaystyle m\text{ and }n$ are non-negative integers, what can be said about their possible values?
5. ## Re: Venn Diagram Problem
Originally Posted by Plato
Surely you can see that $\displaystyle 60-m+n=50$.
If both $\displaystyle m\text{ and }n$ are non-negative integers, what can be said about their possible values?
I don't get it.....
6. ## Re: Venn Diagram Problem
Originally Posted by FailInMaths
I don't get it.....
You know that $\displaystyle -m+n = -10$. As you only have one equation and two unknowns you won't be able to get just one pair of values - but the question isn't asking you that. It's asking you questions about certain values.
For example, is it possible for all 50 units to subscribe to one, the other or both?
,
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### how can we find greatest and smallest possible value of venn diagram
Click on a term to search for related topics.
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0
# finding rectangular equations from parametric equations
how do you find the rectangular equation for the plane curve defined by the parametric equations: x=t-3 and y=t^2+5?
### 2 Answers by Expert Tutors
Jim L. | Dr. Jim - Harvard graduate tutor - perfect SAT score!Dr. Jim - Harvard graduate tutor - perfe...
5.0 5.0 (170 lesson ratings) (170)
1
Hi Morgan,
Converting parametric equation to a cartesian equation or rectangular form involves solving for t in terms of x and then plugging this into the y equation. Therefore,
t=x+3
y = (x+3)^2 + 5
y = x^2+6x + 9 + 5
y= x^2 + 6x + 14
Hope that helps. Jim
Isaak B. | Good (H.S. or College Math, Physics, Chem, EE Engineering) Cheap TutorGood (H.S. or College Math, Physics, Che...
4.9 4.9 (910 lesson ratings) (910)
0
Yes, Jim is entirely correct.
I would only augment his answer by pointing out that in general you need to manipulate the parametric equations to eliminate the parametric variable.
It made the most sense to eliminate t by solving the first equation for t, in this particular problem, because that equation was linear so each value of x only corresponded to one value of t. Things would have gotten a bit messy in this case if you had tried to solve the second equation for t, in this problem, but in general you eliminate the non rectangular-grid parameter by whatever means possible if you want the expression converted to its rectilinear coordinates.
Which equation should be solved for the parametric variable depends on the problem -- whichever equation can be most easily solved for that parametric variable is typically the best choice. For instance had the problem been y = t -3, and x = t^2 + 5, I hope you see that solving for t in terms of y would make more sense, for exactly the same reasons already discussed.
So you solve for t in terms of x if that's easier, or solve for t in terms of y if that's easier, then put the result into the other equation. In this problem, the former was easier.
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Re: Fibonacci string
John W. Krahn <someone@xxxxxxxxxxx> wrote:
> John W. Krahn wrote:
>> David K. Wall wrote:
>>
>>> Here's a fun fact I ran across in the book _The Golden Ratio_,
>>> by Mario Livio. Take the string '1' and replace it with '10'.
>>> Thereafter, replace any occurrence of '1' with '10' and '0' with
>>> '1'. Then count the number of 0s and 1s in the string. You get
>>> a Fibonacci sequence for each count (offset by one iteration).
>>>
>>> Here's Perl code to do it. You get about the same results if
>>> you start with '0', just offset a little.
>>>
>>> use strict;
>>> use warnings;
>>>
>>> my \$v = '1';
>>> for (1 .. 20) {
>>> my (\$n0, \$n1) = (0, 0);
>>> \$v = join '',
>>> map {
>>> if (\$_) {
>>> \$n1++;
>>> '10';
>>> }
>>> else {
>>> \$n0++;
>>> '1';
>>> }
>>> } split //, \$v;
>>> printf "%10d %10d\n", \$n0, \$n1;
>>> }
>>
>> You can make that shorter and faster:
>>
>> my \$v = '1';
>> for ( 1 .. 20 ) {
>> printf "%10d %10d\n", \$v =~ y/0//, \$v =~ y/1//;
>> \$v =~ s/([01])/ \$1 ? '10' : '1' /eg;
>> }
>
> A bit faster. :-)
>
> my \$v = '1';
> for ( 1 .. 20 ) {
> printf "%10d %10d\n", \$v =~ y/0//, \$v =~ y/1//;
> \$v =~ s/./ \$& ? '10' : '1' /eg;
>}
I bow to greater perl-fu, but with the caveat that I was more
interested in making the idea clear than in coming up with a fast
implementation. I'd accuse you of having too much time on your hands
if I weren't guilty of the same sort of thing myself.
.
Relevant Pages
• Re: replacement of slow unpack
... The lesson here is ...
(comp.lang.perl.misc)
• Re: Fibonacci string
... John W. Krahn wrote: ... Then count the number of 0s and 1s in the string. ... Here's Perl code to do it. ... You get about the same results if you start with '0', just offset a little. ...
(comp.lang.perl.misc)
• Re: Delete characters
... John W. Krahn wrote: ... >> I need to delete the first five characters of a string. ...
(comp.lang.perl.misc)
• Re: before -
... John W. Krahn wrote: ... my \$string = "75664545-bookings"; ... Regards ...
(perl.beginners)
• Re: RegExp Problem using Substitutions.
... John W. Krahn wrote: ... as a result I will get a string "aaassass". ... (Using only one substition.)) ...
(perl.beginners)
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# Sign Of A Cancer
The Life Course number is one of the most important since it is your main purpose in life and the primary reason you were birthed; sign of a cancer the expression number is how you reveal on your own to the world and exactly how others see you.
The soul’s desire number makes you pleased in life and what brings you happiness, and the birthday celebration number is all the skills and skills and abilities that you come down into this lifetime with to sustain your life’s function.
So I’m mosting likely to reveal you exactly how to compute the Life Course number initially because that is the most critical number in your chart, and it is your primary function in life, so the Life Course number is established by your day of birth.
So all we’re doing is we are simply adding every one of the numbers with each other in your date of birth till we break them down and obtain a single figure.
So in this example, we have December 14, 1995.
So we require to get a single figure for each and every area.
First, we need to obtain a solitary digit for the month.
We need to obtain a solitary figure for the day of the month, and we need to get a solitary number for the birth year, and afterwards when we obtain a single digit for each one of these, we can add these 3 together.
So December is the 12th month of the year, so we require to add the one in both to get a solitary number.
One plus two equates to three, and for the day of the month, we need to add the one in the four with each other, which will certainly provide us number five for the day of the month.
And then we need to include 1995 with each other.
Which will provide us a solitary figure for the year, so one plus 9 plus 9 plus 5 equals twenty-four, and after that we require to add the 2 in the four with each other since we have to simplify to one number for the year.
So 2 plus 4 amounts to six.
So when we have a single figure each of these areas, a single number for the month, a single figure for the day, and a solitary number for the year, after that we’re mosting likely to add the three of these together, and in this example.
It would be three plus five plus six equals 14, and afterwards we have to add the one and the four together due to the fact that we’re attempting to damage every one of this down till we obtain a single figure.
So if you include the one in the 4 with each other in 14, you obtain a number 5.
So in this scenario, this individual’s life course number is a number five.
He or she is a 14 5, and it’s necessary to bear in mind of the 2 last numbers that we totaled to get the Life Course number 5 due to the fact that these 2 last numbers are important to the individual’s life course number.
So, when it comes to a Life Path, the number 5 can be a fourteen five since the four equals number 5.
They might likewise be a two and a 3 a twenty-three number 5. Still, those two numbers that you included together to obtain that Life Course number are significant since they tell you what energies you will require to utilize throughout your lifetime to accomplish your life function. In this situation, sign of a cancer, this person has a Life Course number of number 5.
Still, to accomplish their life path, the variety of a number five, they are going to require to utilize the power of the number one and the number 4 to achieve their function, so take note of those 2 last numbers that you totaled to obtain your last Life Path number due to the fact that those are really vital.
A fourteen-five is mosting likely to be very various than a twenty-three-five.
If you have any questions concerning these two last numbers used to acquire your final Life Path, number remark listed below, and I will try to address your questions now.
I wanted to reveal you this example since, in some situations, we do not damage down all of the numbers to get a solitary digit.
So in this instance, we have December 14, 1992, and when we added every one of the numbers together in the month, the day, and the year, we finished up with a number 11 and 11 in numerology is a master number and the master numbers.
We do not include the 2 numbers with each other, so there are 3 master numbers in numerology, and the three master numbers are 11, 22, and 33.
So after you have actually included every one of the digits with each other in your birth date and if you wind up with either an 11, a 22, or a 33, you will certainly not include these 2 digits with each other because you have a master number Life Course.
Number and the master numbers are various from the various other numbers in numerology since they hold the dual numbers’ power, and we do not add both figures together in these scenarios.
So if you have either an 11, a 22, or 33, you will certainly not add both numbers together.
You will maintain it as is, and you have a master number as a life path.
In this circumstance, the number that we combined to obtain the 11 were 8 and 3.
Those are significant numbers in this scenario due to the fact that this master number 11 will need to utilize the eight and the 3.
To get their number 11 life objective, so in 83, 11 will be a great deal various from on 92 11 because an 83 11 will need to use the power of the 8 in the 3 to get their life function. The 9211 will Need to use the nine and both indicate receive their 11 life function.
So the birthday number is probably the most accessible number to calculate in your graph due to the fact that for this number, all you have to do is add the numbers together of the day you were born upon, so he or she was birthed on December 14, 1995.
So we will include the one and the four with each other due to the fact that those are the numbers of the day. sign of a cancer
He or she was birthed, so 1 plus 4 amounts to 5.
So he or she’s birthday celebration number is a number 5.
Now, in this example, December 11, 1995.
He or she was born on the 11th day of the month, and 11 is a master number, so we do not include both ones with each other since 11 is a master number, and there are 3 master numbers in numerology, 11, 22, and 33.
So if you were born upon the 11th of a month or the 22nd of a month, you would certainly not include both figures with each other because your birthday celebration number is a master number.
So your birthday number is either a master number 11 or a master number 22. The master numbers are the only numbers in numerology that we do not add both numbers together to get a final number.
So you will certainly keep those two numbers alone, and you will certainly not include them with each other.
For the last 2 numbers, you have utilized your date of birth to compute those 2 numbers, but for the expression number and the heart’s urge number, you will certainly utilize the complete name on your birth certificate.
You’re going to utilize your first, center, and last name on your birth certification to calculate your expression and your heart’s desire number, and so we’re now mosting likely to use the Pythagorean number system to calculate these numbers.
And it’s called the Pythagorean system because Pythagoras, a Greek mathematician, developed it. He was the mathematician that developed the Pythagorean theory. He is the papa of numerology, and he discovered that all numbers hold energy. They all possess a Certain vibration, therefore after discovering out that all numbers hold particular energy, Pythagoras developed the Pythagorean number system. From that, we have contemporary numerology today.
It is a chart with all of the letters of the alphabet and all letters matching to a specific number.
So essentially, all letters in the alphabet have the energy of a number.
And if you consider this chart, you can find out what number each letter has the energy of.
So a has the power of a top B has the power of a number.
2 C has the power of a number 3 and so on, and so forth.
So, for the expression number, all you have to do is add every one of the letters in your full birth name, so the first, center, and last name on your birth certificate and reduce them down to one digit. sign of a cancer
So in this instance, we have Elvis Presley.
So what I did was I checked out the number graph, and I found the equivalent number per letter in Elvis’s name.
So E is a 5, l is a 3 V is a 4.
I am a 9, and I just found all of the numbers representing every one of the letters in his name; and I included every one of those numbers together, and the last number I got was an 81.
Then I included the 8 and the one together due to the fact that we have to maintain damaging these down up until we obtain a single-digit, and when I stated the eight and the one together, I got a nine, so Elvis’s expression number is a nine.
Now the one circumstance where you would not proceed to include these numbers together until you got a single digit would be if you obtained a master number, so the 3 master numbers are 11, 22, and 33.
If you got a master number, you would certainly not remain to add these numbers with each other.
You will keep them as either 11, 22, or 33.
So it coincides scenario as it was with the life fifty percent number and the birthday number.
The master numbers are unique, and we do not add the 2 figures with each other, so proceeding to the soles prompt number so often the soles motivate number can be called the sole number, and it can likewise be called the heart’s desire number.
These words are used mutually however recognize whenever you see the single number or heart’s wish number that they are basically the very same thing as a soles prompt number.
For this number, we will include every one of the vowels in your full birth name.
Every one of the vowels in your initial, middle, and surname on your birth certification, and we’re mosting likely to decrease them to one digit.
So we’re mosting likely to use the same Pythagorean chart that we did in the past, and we’re going to look up all of the numbers that represent the vowels in your initial, center, and last name.
So right here we have Kate Middleton.
I lately just did a video clip on her numerology, so I figured why not utilize her today.
So her first middle and last name are Catherine, Elizabeth Middleton, so I looked up the numbers corresponding to only the vowels in her name.
So, as you can see, an amounts to 1 B equals 5, I equals nine, and E amounts to 5, and after that I did that for all of the vowels in her full name.
And after that I simply included all of those numbers together, and that gave me 60, and after that 60 reduces to number 6 due to the fact that we remember we’re just trying to break these numbers down till we obtain a single number.
So in this circumstance, Kate’s Seoul’s desire number is a number 6.
Now you will certainly not remain to damage down the numbers if you get an 11, a 22, or a 33, so, as I said with all the various other numbers if you get among these numbers, this is a master number, and we do not add the 2 figures with each other so 11, 22 and 33 are master numbers.
Sign Of A Cancer
If you get among those as your hearts urge numbers, you will certainly not proceed to add the numbers together; you will keep them as double digits.
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# Partial Fractions
• Jun 11th 2009, 08:39 AM
coobe
Partial Fractions
hi again!
i have to make partialfractions out of the following integral:
$\displaystyle \int \frac{1-x}{x^3-2x^2+x-2} dx$
guessing a critical point for the denominator gives me $\displaystyle x_{0}=2$
then polynomial division gives me
$\displaystyle (x^3-2x^2+x-2) / (x-2) = x^2+1$
$\displaystyle \rightarrow \frac {1-x}{(x-2)(x^2+1)} = \frac {A}{x-2} + \frac{B}{x^2+1}$
is this correct ? if so, i dont seem to get an equation system that is solvable...
could you give me a quick hint to do partial fractions in general ? like the different cases etc..., my professors script is a mess
thanks
• Jun 11th 2009, 09:28 AM
alexmahone
$\displaystyle \frac {1-x}{(x-2)(x^2+1)} = \frac {A}{x-2} + \frac{B}{x^2+1}$
Multiply throughout by (x-2)
$\displaystyle \frac {1-x}{x^2+1} = A + B\frac{x-2}{x^2+1}$
Sub x=2
$\displaystyle -\frac{1}{5}=A$
$\displaystyle A=-\frac{1}{5}$
• Jun 11th 2009, 09:32 AM
AMI
Quote:
Originally Posted by coobe
$\displaystyle \rightarrow \frac {1-x}{(x-2)(x^2+1)} = \frac {A}{x-2} + \frac{B}{x^2+1}$
Actually, it is $\displaystyle \frac {1-x}{(x-2)(x^2+1)} = \frac {A}{x-2} + \frac{Bx+C}{x^2+1}$
$\displaystyle \Rightarrow 1-x=A(x^2+1)+(Bx+C)(x-2)$ $\displaystyle \Rightarrow A+B=0,-2B+C=-1,A-2C=1$ $\displaystyle \Rightarrow A=-\frac{1}{5},B=\frac{1}{5},C=-\frac{3}{5}$.
Did you look here: Partial fraction - Wikipedia, the free encyclopedia ?! It seems pretty well written..
• Jun 12th 2009, 02:43 AM
coobe
sorry i dont get it, why you need a C when there are only 2 factors in the denominator.... and i dont get the x with the B neither :(
• Jun 12th 2009, 02:59 AM
HallsofIvy
Quote:
Originally Posted by coobe
sorry i dont get it, why you need a C when there are only 2 factors in the denominator.... and i dont get the x with the B neither :(
What do you mean by "two factors in the denominator"? The reason we need Bx+ C over $\displaystyle x^2+ 1$ is because it can't be factored!
• Jun 12th 2009, 11:33 AM
AMI
Quote:
Originally Posted by coobe
sorry i dont get it, why you need a C when there are only 2 factors in the denominator.... and i dont get the x with the B neither :(
(Wondering) Did you look at the link?? It's the case where you have an irreducible second degree factor in the denominator, just like HallsofIvy said..
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# What is the maximum value of $n$ with average value must be an integer?
Let $$M$$ be a positive integer greater than $$1$$. All integers from $$1$$ to $$M$$ were written on a board.
Each time we erase a positive integer on the board in a way that the average value of all numbers that have been erased must always be an integer.
Assume that there are $$n$$ numbers that have been erased ($$1 \leq n \leq M$$, $$n$$ is not a constant number). The process will end with $$n$$ numbers if and only if it is impossible to erase the $$(n+1)th$$ number so that the average value of $$n+1$$ erased numbers can be an integer.
For all possible ways to erase the numbers, what is the maximum and the minimum value that $$n$$ can reach?
For example, with $$M=3$$, we have the maximum of $$n$$ is $$3$$ (choose $$a_1=1$$, $$a_2=3$$, $$a_3=2$$ ) , the minimum value of $$n$$ is $$1$$ (choose $$a_1=2$$, then it is impossible to choose $$a_2=1$$ or $$a_2=3$$ because $$\frac{2+1}{2}, \frac{2+3}{2}$$ are not integers). For larger $$n$$, I thought that I can solve with Chinese Remainder Theorem, but I didn't know how to use it.
Is it possible to find the minimum or maximum value of $$n$$?. If not, what are the conditions of $$M$$ so that the minimum or maximum value of $$n$$ can be found?
(Sorry, English is my second language, so the questions may unclear for some readers)
EDIT: I've edited the post because at first I did average as $$\frac{1}{2}\sum a_i$$ instead of $$\frac{1}{i}\sum a_i$$. Below is the corrected answer:
Modeling the problem for maximum or minimum we get:
\begin{align*} \text{max/min }&\sum_{i=1}^{m}b_i\\ \text{such that }& b_11+b_22+\cdots+b_MM=Mk\\ &b_i\in\{0,1\},k\in\mathbb{N} \end{align*}
Suppose $$M=1$$. We can erase the only value: $$1$$, and it's average is $$\frac{1}{1}=1$$ an integer. Therefore max=min=$$1$$.
Suppose $$M=2$$. We can only erase $$2$$, because $$\frac{3}{2}\notin \mathbb{Z}$$. Therefore max=min=$$1$$.
Now let's suppose $$M\geq 3$$. Let's see for what $$M$$'s we can erase all numbers, that is $$n=M$$: $$1+2+\cdots+M = \frac{M(M+1)}{2} = Mk \rightarrow M=2k-1$$ Hence for $$M\in\{3,5,7,9,\cdots\}$$ we can erase all numbers to get it's average as an integer.
We've covered the cases for $$M$$ when it's odd and $$M\geq 3$$. Let's see what happens for the cases where $$M$$ is even, that is, $$M=2q$$. $$1+2+\cdots+2q = \frac{2q(2q+1)}{2} = q(2q+1) = 2q^2+q$$ We want that result to be equal to $$Mk=2qk$$ for some $$k \in \mathbb{N}$$. Therefore if we let $$b_q=0$$ we get the sum as: $$1+2+\cdots + 2q - q = 2q^2+q - q = 2q^2$$ And that is equal to $$2qk$$ for $$k=q$$. Therefore for $$M=2q$$ we get $$n=M-1$$.
That means that when $$M=2q$$ we only need $$b_q=0$$ to guarantee that the average of the sum of the erased numbers is an integer.
Finally see that for $$M\geq 1$$ you can always use the same reasoning for $$M=1$$ for the minimum... You remove $$1$$ and the average of the removed numbers is going to be $$\frac{1}{1}=1$$ that is an integer.
Since we've covered all cases for $$M$$, we're done.
Examples: For $$M=12345$$: $$1 + 2 + \cdots + 12345 = 76205685 \text{ and } \frac{76205685}{12345}=6173$$ For $$M=124=2\cdot 62$$: $$1 + 2 + \cdots + 124 - 62 = 7750 - 62 = 7688 \text{ and } \frac{7688}{124}=62$$
• Thanks for your answer. However, my question is that the average value, means the total sum of numbers divided by the amount of numbers, ($\frac{a_1+a_2+...+a_i}{i}$ is always an integer, not half of the sum. – apple Oct 26 '18 at 15:22
• @apple Oh mate, sorry!!! I'll try to correct it. – Bruno Reis Oct 26 '18 at 15:28
• @apple I've got the result. I just don't have time to post it now. I'll try to post it within 3 hours! – Bruno Reis Oct 26 '18 at 15:42
• @apple It's done! See if you can get it. – Bruno Reis Oct 26 '18 at 18:21
• @Brunu: I don't think you understood the problem correctly. For odd $M$ you only prove that you can get the last sum correctly, that is (comperatively) easy. The problem is to select one number $a_i$ after the other, and after each step have an averge that is an integer. So for $M=5$, the first chosen number must be divisible by $1$ (easy), the sum of the first and second must be divisible by $2$, the sum of the first $3$ must be divisible by $3$, a.s.o, as far as you can get. Reread the description again and I think you will see what you misunderstood. For $M=5$, the max $n$ is e.g. 3. – Ingix Oct 26 '18 at 21:33
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# What is the value of x which satisfies 2^1995+2^1995+2^1995+2^1995+2^1995+2^1995++2^1995+2^1995=2^x a.1996 b.1997 c.1998 d.1999 e.2000
`2^1995+2^1995+2^1995+2^1995+2^1995+2^1995+2^1995+2^1995=2^x`
We can simplify this problem as:
`8(2^1995)=2^x`
We know that 8=2^3 theregore:
`(2^3)(2^1995)=2^x`
Following the rule of exponenets that state the following:
`x^mx^n=x^(m+n)`
Therefore:
`(2^3)(2^1995)=2^(3+1995)=2^1998=2^x`
Therefore, x=1998, so the answer is c.
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## Monday Math 30
We can find the definite integral $\int_0^{\infty}\frac{\sin^2x}{x^2}\,dx$ in three different ways:
1. Long: Differentiation under the integral sign.
In my previous post on this topic, I showed that $\int_{0}^{\infty}\frac{\sin{x}}{x}\,dx=\frac{\pi}{2}$. We can use a similar method on $\int_{0}^{\infty}\frac{\sin{kx}}{x}\,dx$, namely, defining $f(a)=\int_{0}^{\infty}e^{-ax}\frac{\sin{kx}}{x}\,dx$, and then finding
$\begin{array}{rcl}f'(a)&=&\int_{0}^{\infty}\frac{\partial}{\partial{a}}\left(e^{-ax}\frac{\sin{kx}}{x}\right)\,dx\\&=&-\int_{0}^{\infty}e^{-ax}\sin{kx}\,dx\\&=&\left[\frac{e^{-ax}(k\cos{kx}+a\sin{kx})}{a^2+k^2}\right]_{x=0}^{\infty}\\&=&-\frac{k}{a^2+k^2}\end{array}$.
For k=0, this gives $f(a)=0$, as expected.
For non-zero k, we integrate with respect to a, and find
$f(a)=C-\arctan\left(\frac{a}{k}\right)$.
Again, we note that as $a\to\infty$, we see that for all x>0, $e^{-ax}\frac{\sin{kx}}{x}\to0$, and thus $\lim_{a\to\infty}f(a)=0$.
For k>0, we have $\lim_{a\to\infty}\arctan\left(\frac{a}{k}\right)=\frac{\pi}{2}$, giving $C=\frac{\pi}{2}$, and so $f(a)=\frac{\pi}{2}-\arctan\left(\frac{a}{k}\right)$.
For k<0, we instead have $\lim_{a\to\infty}\arctan\left(\frac{a}{k}\right)=-\frac{\pi}{2}$, giving $C=-\frac{\pi}{2}$, and thus $f(a)=-\frac{\pi}{2}-\arctan\left(\frac{a}{k}\right)$.
Thus $\int_{0}^{\infty}\frac{\sin{kx}}{x}\,dx=f(0)= \begin{cases}\frac{\pi}{2},&\;k>{0}\\{0},&\;k={0}\\-\frac{\pi}{2},&\;k<{0}\end{cases}$.
Now, to our original integral $\int_0^{\infty}\frac{\sin^2x}{x^2}\,dx$.
Define $I(p)=\int_0^{\infty}\frac{\sin^2(px)}{x^2}\,dx$, for p≥0
Then
$\begin{array}{rcl}I'(p)&=&\int_0^{\infty}\frac{\partial}{\partial{p}}\left(\frac{\sin^2(px)}{x^2}\right)\,dx\\&=&\int_0^{\infty}\frac{2\sin(px)\cos(px)}{x}\,dx\\&=&\int_0^{\infty}\frac{\sin(2px)}{x}\,dx\end{array}$,
and now we know, from our previous work, that for p>0, this is $\frac{\pi}{2}$. Integrating, $I(p)=\frac{\pi}{2}p+C$, and as $I(0)=0$, we see $I(p)=\frac{\pi}{2}p$, and so our integral is
$\int_0^{\infty}\frac{\sin^2x}{x^2}\,dx=I(1)=\frac{\pi}{2}$.
2. Difficult: Trig and Double Integral.
We can use the trigonometric identity $\sin^2x=\frac{1-\cos(2x)}{2}$ to rewrite our integral:
$\int_0^{\infty}\frac{\sin^2x}{x^2}\,dx=\int_0^{\infty}\frac{1-\cos(2x)}{2x^2}\,dx$.
Substituting $u=2x$, we have
$\int_0^{\infty}\frac{1-\cos(2x)}{2x^2}\,dx=\int_0^{\infty}\frac{1-\cos{u}}{\frac{u^2}{2}}\,\frac{du}{2}=\int_0^{\infty}\frac{1-\cos{u}}{u^2}\,du$
Now, here is the hard part: we can recognize that $\int_0^{\infty}xe^{-kx}\,dx=\frac{1}{k^2}$. Thus, $\frac{1}{u^2}=\int_0^{\infty}ve^{-uv}\,dv$, and our integral can be rewritten as the double integral
$\int_0^{\infty}\frac{1-\cos{u}}{u^2}\,du=\int_0^{\infty}\int_0^{\infty}(1-\cos{u})ve^{-uv}\,dv\,du$.
We can reverse order of integration to solve this:
$\begin{array}{rcl}\int_0^{\infty}\frac{1-\cos{u}}{u^2}\,du&=&\int_0^{\infty}\int_0^{\infty}(1-\cos{u})ve^{-uv}\,du\,dv\\&=&\int_0^{\infty}\left[\left(\frac{v^2\cos{u}}{v^2+1}-\frac{v\sin{u}}{v^2+1}-1\right)e^{-uv}\right]_{u=0}^{\infty}\,dv\\&=&\int_0^{\infty}\frac{1}{1+v^2}\,dv\\&=&\left[\arctan{v}\right]_{v=0}^{\infty}\\&=&\frac{\pi}{2}\end{array}$
3. Simple (but needing some background): Fourier Transform.
Defining the sine cardinal as $\mathrm{sinc}(x)=\frac{\sin(\pi{x})}{\pi{x}}$. It is known that the Fourier transform transforms between the sinc function and the rectangle function: namely, $\mathcal{F}(\mathrm{sinc}(t))(f)={\Pi}(f)$, where the function Π(x) is the rectangle function ${\Pi}(x)=\begin{cases}1,&|x|\le{\normalsize\frac{1}{2}}\\{0},&|x|>{\normalsize\frac{1}{2}}\end{cases}$
The convolution theorem tells us that:
$\begin{array}{rcl}\mathcal{F}(\mathrm{sinc}^2(t))(f)&=&\mathcal{F}(\mathrm{sinc}(t)\cdot\mathrm{sinc}(t))(f)\\&=&\mathcal{F}(\mathrm{sinc}(t))(f)*\mathcal{F}(\mathrm{sinc}(t))(f)\\&=&{\Pi}(f)*{\Pi}(f)\end{array}$,
where the * represents convolution. The convolution of the rectangle function with itself is the triangle function:
${\Pi}(x)*{\Pi}(x)=\Lambda(x)\equiv\begin{cases}{0},&|x|\ge{1}\\1-|x|,&|x|<1\end{cases}$.
And so $\mathcal{F}(\mathrm{sinc}^2(t))(f)={\Lambda}(f)$,
Writing this out with the definition of the Fourier transform, we have:
$\int_{-\infty}^{\infty}\mathrm{sinc}^2(t)e^{-2\pi{i}ft}\,dt={\Lambda}(f)$.
Plugging in f=0, and using Λ(0)=1,
$\int_{-\infty}^{\infty}\mathrm{sinc}^2(t)\,dt=1$.
The sinc function is even, so we have
$\int_{0}^{\infty}\mathrm{sinc}^2(t)\,dt=\frac{1}{2}$
Using the subsistution xt, we have
$\begin{array}{rcl}\int_{0}^{\infty}\frac{\sin^2(\pi{t})}{(\pi{t})^2}\,dt&=&\frac{1}{2}\\\int_{0}^{\infty}\frac{\sin^2{x}}{x^2}\,\frac{dx}{\pi}&=&\frac{1}{2}\\\int_{0}^{\infty}\frac{\sin^2{x}}{x^2}\,dx&=&\frac{\pi}{2}\end{array}$
QED.
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# What Is 30 Multiplied By 80?
30 * 80 = 2400
### What is the Answer to the following Calculation: 30 x 80
In this section, we'll address a single issue: What is the product of 30 multiplied by 80? (Or, alternatively, what is 80 times 30?) The following is the response: 80 times 30 is equal to 2400.
Here are some alternative ways to show or express the fact that 30 x 80 = 2400
### Eighty times Thirty equals Two Thousand Four Hundred
Here are some alternative ways to show or express the fact that 30 x 80 = 2400
• Eighty times Thirty equals Two Thousand Four Hundred
• 30 (80) = 2400
Consider the sum of 30 multiplied by 80 to understand better what "80 times 30" means. Add the Eighty digits together to get the result, or you may write down the number 30 80 times.
When using a handheld calculator, you can double-check the solution to 2400 by pressing 30 x 80 and then = to get the result.
The answer to this riddle is 30 x 80 is 2400.
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## Featured Calculations
### In what ways is it useful to multiply numbers?
Finding the total number of things requires some multiplication. To do this, we'll consider the total number of items and the number of groups of similar size.
### What do we name the numbers used in multiplication?
Typically, the "factors" are the input integers into a multiplication problem. The "multiplicand" refers to the number that is multiplied, whereas the "multiplier" refers to the number that is used to do the multiplication.
### What is the inverse of multiplication?
Subtraction has an inverse operation for multiplication. Addition can be cancelled out by subtraction. Adding and subtracting are polar opposites in the mathematical world.
### How many multiplication facts are there?
Since integers can be multiplied forever, there are an unlimited number of multiplication tables. Typically, students study 144 of these multiplication facts, ranging from 1 to 12. A times table or multiplication chart is a common way to display data using multiplication.
### What are the 4 rules of multiplication?
1. Multiplying 0 by any number always results in 0.
2. Any number times one is always the same number.
3. When multiplying by 10, add a zero to the first digit of the original integer.
4. When multiplying by two numbers that have the same sign, the result is always a positive number.
### How many properties of multiplication are there?
There are three properties of multiplication:
1. Commutative
2. Associative
3. Distributive
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# How Do Percentage Points Work?
When there’s an election, the newspapers often write about the amount of support all the different parties have in an effort to give the readers an image of the political landscape. They might report that one party has the support of $30\phantom{\rule{0.17em}{0ex}}\text{%}$ of the voters, and that this is an increase by $2.3$ percentage points compared to last month. When this percentage increases or decreases in the polls, you’re not given those changes in percentage change, but a change in percentage points. This works the same way as adding or subtracting with normal numbers, meaning that if a party with $30\phantom{\rule{0.17em}{0ex}}\text{%}$ support gains $2.3$ percentage points, they now have $30\phantom{\rule{0.17em}{0ex}}\text{%}+2.3\phantom{\rule{0.17em}{0ex}}\text{%}=32.3\phantom{\rule{0.17em}{0ex}}\text{%}$ support.
Theory
### PercentagePoints
Percentage points describe the change of a number given in percentages, where the change follows the normal rules of adding and subtracting.
Example 1
A party becomes more popular in the polls, going from $4\phantom{\rule{0.17em}{0ex}}\text{%}$ to $6\phantom{\rule{0.17em}{0ex}}\text{%}$. That means the support of that party has increased by
$6-4=2$
percentage points. If you want to find the percentage change in support, you’ll find it to be $50\phantom{\rule{0.17em}{0ex}}\text{%}$, because
$\frac{6-4}{4}=\frac{2}{4}=0.5=50\phantom{\rule{0.17em}{0ex}}\text{%}.$
Example 2
In a survey, $\text{}30.1\text{}\phantom{\rule{0.17em}{0ex}}\text{%}$ of the population of Washington D.C. answers that they have used public transport as part of their commute at least once a week. Two years later, this has increased to $\text{}35.7\text{}\phantom{\rule{0.17em}{0ex}}\text{%}$. That means that the number of people who use public transport as part of their commute has increased by
$\text{}35.7\text{}-\text{}30.1\text{}=\text{}5.6\text{}$
percentage points in two years. What’s the growth in percentages?
Now you need to use the formula for percentage change:
You can see that the percentage change is $18.6\phantom{\rule{0.17em}{0ex}}\text{%}$, while the change in percentage points is $5.6$.
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# Why both data have same standard deviation?
But I am curious why the standard deviation for both data is same. I think because data 2 is "100 - data 1".
• $var(X+a)=var(X)$ and $var(aX)=a^2var(X)$ for a real number $a$ and a random variable $X$. With those, can you prove why $var(100-X)=var(X)$ where X is your data set? (Equal variances is the same as equal standard deviations.)
– Dave
Commented Feb 15, 2020 at 0:24
I still would like you to try to prove it, but I’ll give some intuition about those two identities I gave.
Remember that variance (and standard deviation) have something to do with how spread out your data are.
$$var(X+a)=var(X)$$ means that if you take data with some amount of spread and slide them up or down the number line, you do not change the spread.
$$var(X)=var(-X)$$ is not quite what I gave, but it’ll be part of your proof. The intuition here is that you’re just spinning it around, taking a mirror image. The data are as spread out as their reflection.
Together, $$var(100-X)=var(X)$$ means that you flip the data to the mirror image and then slide along the number line, but you do not change the spread.
Capisce?
• what is the meaning of Capisce?
– aan
Commented Feb 16, 2020 at 11:41
• that's Italian for "do you comprehend ?" Commented Feb 16, 2020 at 13:20
I believe the answer to this question can be expressed in more elemental terms. Beginning with the formula for the sample standard deviation
$$s_{x} = \sqrt{\frac{ \sum_{i=1}^{n}{(x_i - \overline{x})^2} }{n - 1}},$$
I want to draw your attention to the expression $$(x_i - \overline{x})$$ in the numerator. Ignore the summation notation for the purposes of this answer. Note, we decrement the sample mean (i.e., $$\overline{x}$$) from each realization of $$x_{i}$$. If we were to increase or decrease each $$x_{i}$$ by the same amount, the mean will change. However, the distance of each realization of $$x_{i}$$ from that central tendency remains the same. In other words, each deviation from the mean is the same.
Subtracting each $$x_{i}$$ from a constant (e.g., $$100 - x_{i}$$) flips the vector of values to its mirror image, then slides it along the number line by a constant amount. Suppose the first realization of $$x_{1}$$ = 20 and $$\overline{x}$$ = 4. In keeping with the foregoing expression,
$$(20 - 4) = 16,$$
which is the deviation from the sample mean. Now, assume we don't move along the number line just yet and we simply put a negative sign in front of each $$x_{i}$$, such that we have $$-(x_{i})$$; this flips the sign of the mean as well. The first realization, $$x_{1}$$, is now $$-20$$. Again, substituting $$-20$$ into the expression,
$$(-20-(-4)) =-16,$$
which is the same number of units away from the mean. Note, squared deviations result in a positive number, so the numerator remains the same. You can run this code in R which builds upon @knrumsey's insightful response, but breaks it down further:
x <- rnorm(10000, 30, 5) # Simulates 10,000 random deviates from the normal distribution
par(mfrow = c(2, 2))
hist(x, xlim = c(0, 100), col = "blue")
hist(-x, xlim = c(-100, 0), col = "red") # Note the -x, it simply flips the sign (mirror image)
hist(x, xlim = c(0, 100), col = "blue")
hist(100-x, xlim = c(0, 100), col = "red") # Subtracting x from 100 shifts values along the x-axis
The first row of plots shows how the realizations translate when we negate each $$x_{i}$$. The second row shows what happens when each $$x_{i}$$ is subtracted from 100; the translation first flips then slides across the number line. The spread of the distribution is unaffected by this.
To supplement @Dave's answer, take a look at the following histograms which have the same x-axis. Data2 is just a shifted version of Data1 and therefore the standard deviation, which is a measure of spread, shouldn't change.
R code to generate histograms:
x <- 100*rgamma(1000, 6, 2) #Simulate some data
hist(x, xlim=c(0,100))
hist(100-x, xlim=c(0,100))
• Thanks. well-plotted histogram. May I know how do you plot the histogram?
– aan
Commented Feb 15, 2020 at 5:29
• @aan see the update. Commented Feb 15, 2020 at 17:49
• @knrumsey Good illustration. However, simulating from the gamma distribution would produce realizations that cluster around a mean that is much lower than what I see here given the parameters you specified. Right? Commented Feb 16, 2020 at 20:24
• @Tom, that is correct. I didn't put too much thought into which distribution I simulated from. Just wanted draws between 0 and 100. Commented Feb 16, 2020 at 20:45
• @Thomas Bilach, realizing that I misunderstood your question. Thanks for pointing this out, I've updated the post to reflect the fact that I multiplied the data by 100. Commented May 3, 2020 at 0:48
changes period to period in DATA 1 identical to the negative changes in DATA 2
OBS DATA 1 FIRST DIF DATA 2 FIRST DIFF
1 30 NA 70 NA
2 40 10 60 -10
3 80 40 20 -40
4 30 -50 70 50
5 20 -10 80 10
6 33 13 67 -13
7 33 0 67 0
8 33 0 67 0
9 33 0 67 0
CONTINUING STEPS TO UNCOVER THE BASIC RELATIONSHIP OF DATA1 AND DATA2:
I took the series DATA1 and computed the ACF ... here it is ..
AND for DATA2 here
...note that not only are the standard deviations the same but the ACF's are the same.
• This is an interesting perspective, but I believe it would need more explanation to be understood by many readers.
– whuber
Commented Feb 15, 2020 at 22:53
• @IrishStat, thanks. do you mind to explain more?
– aan
Commented Feb 16, 2020 at 11:42
• i added some material to my response detailing how differenced data period to period ( val2-val1,val3-val2,.. val9-val8) for both original series DATA1 and DATA2 were complements of each other Commented Feb 16, 2020 at 13:19
• @IrishStat. Noted. what is the meaning of Q.E.D and OBS?
– aan
Commented Feb 16, 2020 at 15:58
• An explicit connection between those assertions and the result is needed. What is interesting is your use of successive differences ("changes period to period" in an otherwise unordered dataset) to characterize the standard deviation of the dataset. In other words, you appear to be claiming that the set of equalities $$x_{i+1}-x_i = -(y_{i+1}-y_i)$$ for $i=1, 2, \ldots, n-1$ implies $$\sum_{i=1}^n(x_i-\bar x)^2=\sum_{i=1}^n(y_i-\bar y)^2.$$ That's true, but it does need to be shown: you can't just exhibit a table of numbers and declare "QED"!
– whuber
Commented Feb 16, 2020 at 20:15
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# 2) Take the digits 4,5,6,7,9 29 and make any three8 digit numbers,
2) Take the digits 4,5,6,7,9 29 and make any three
8 digit numbers,
### 2 thoughts on “2) Take the digits 4,5,6,7,9 29 and make any three<br />8 digit numbers,”
Given digits are 4, 5, 6, 7, 8 and 9.
For making three numbers with eight digits we are putting two digits twice as we are having only 6 digits in the question.
Three numbers are
1. 46579868
2. 57649845
3. 78965469
By using commas
1. 4, 65, 79, 868
2. 5, 76, 49, 845
3. 7, 89, 65, 469
Here, we are putting commas based on the Indian Value System.
Step-by-step explanation:
@darksoul
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How many degrees is a regular pentagon?
Understand the Problem
The question is asking for the measure of each interior angle of a regular pentagon, which is a five-sided polygon. To find this, we need to use the formula for calculating the interior angles of a polygon.
108 degrees
The measure of each interior angle of a regular pentagon is 108 degrees
Steps to Solve
1. Identify the formula for the sum of interior angles of a polygon
The formula for the sum of the interior angles of a polygon with $n$ sides is $(n-2) imes 180^\circ$.
1. Calculate the sum of the interior angles for a pentagon
Substitute $n = 5$ into the formula:
$$(5-2) imes 180^\circ = 3 imes 180^\circ$$
Therefore, the sum of the interior angles of a pentagon is $540^\circ$.
1. Calculate the measure of each interior angle in a regular pentagon
Since all interior angles in a regular pentagon are equal, divide the sum of the interior angles by 5:
$$\frac{540^\circ}{5} = 108^\circ$$
The measure of each interior angle of a regular pentagon is 108 degrees
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# Being Thoughtful - Primary Number
### Five Steps to 50
##### Stage: 1 Challenge Level:
Use five steps to count forwards or backwards in 1s or 10s to get to 50. What strategies did you use?
### Ring a Ring of Numbers
##### Stage: 1 Challenge Level:
Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.
### Heads and Feet
##### Stage: 1 Challenge Level:
On a farm there were some hens and sheep. Altogether there were 8 heads and 22 feet. How many hens were there?
### The Tall Tower
##### Stage: 1 Challenge Level:
As you come down the ladders of the Tall Tower you collect useful spells. Which way should you go to collect the most spells?
### More Numbers in the Ring
##### Stage: 1 Challenge Level:
If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?
### Dotty Six
##### Stage: 1 and 2 Challenge Level:
Dotty Six is a simple dice game that you can adapt in many ways.
### Biscuit Decorations
##### Stage: 1 and 2 Challenge Level:
Andrew decorated 20 biscuits to take to a party. He lined them up and put icing on every second biscuit and different decorations on other biscuits. How many biscuits weren't decorated?
### Four Go
##### Stage: 2 Challenge Level:
This challenge is a game for two players. Choose two numbers from the grid and multiply or divide, then mark your answer on the number line. Can you get four in a row before your partner?
### Shape Times Shape
##### Stage: 2 Challenge Level:
These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?
### Play to 37
##### Stage: 2 Challenge Level:
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
### Cows and Sheep
##### Stage: 2 Challenge Level:
Use your logical reasoning to work out how many cows and how many sheep there are in each field.
### Number Differences
##### Stage: 2 Challenge Level:
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?
### Mystery Matrix
##### Stage: 2 Challenge Level:
Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.
### Magic Vs
##### Stage: 2 Challenge Level:
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
### Shapes on the Playground
##### Stage: 2 Challenge Level:
Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?
### Nice or Nasty
##### Stage: 2 and 3 Challenge Level:
There are nasty versions of this dice game but we'll start with the nice ones...
### Factor Lines
##### Stage: 2 and 3 Challenge Level:
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
### Make 37
##### Stage: 2 and 3 Challenge Level:
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
### First Connect Three
##### Stage: 2 and 3 Challenge Level:
The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?
### The Remainders Game
##### Stage: 2 and 3 Challenge Level:
A game that tests your understanding of remainders.
### Dicey Operations
##### Stage: 3 Challenge Level:
Who said that adding, subtracting, multiplying and dividing couldn't be fun?
### Two and Two
##### Stage: 3 Challenge Level:
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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# Tagged Questions
46 views
### Eigenvalues and Eigenvectors Diagonilization
Let $A=\begin{bmatrix} -7 & -1 \\ 12 & 0 \\ \end{bmatrix}$ . Find a matrix $P$ and a diagonal matrix $D$ such that $PDP^{-1} = A$. Ok so the first thing I need to look ...
79 views
### If A is a square matrix, and $A^3$=I, how can I prove that eigenvalues of A are either 1 or -1?
If $A$ is a square matrix, and $A^3$=I, how can I prove that eigenvalues of $A$ are either $1$ or $-1$? Thanks for all your input, I figured it out! :)
56 views
### Example of $5\times 5$ matrix with exactly $2$ distinct eigenvalues
What is an an example of a $5\times 5$ matrix with exactly $2$ distinct eigenvalues? Also, what would be the eigenvalues in the example?
45 views
### Diagonalization, Solving a System of Linear Equations
I have questions regarding the following task, which is to diagonalize the matrix A: $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 9 & -20 \\ 0 & 4 & -9 \end{pmatrix}$ What I have ...
63 views
### Solving for eigenvectors (Introduction to Linear Algebra by Serge Lang, example 2 page 240)
In "Introduction to Linear Algebra" by Serge Lang, example 2 page 240, there is the matrix $\begin{pmatrix}1&4\\2&3\end{pmatrix}$. The characteristic polynomial is $(t-5)(t+1)$. The system to ...
62 views
### A quesion in Fulton & Harris book “representation theory a first course”
In Section 11.2 A little plethysm, it discusses the tensor product of two different representations of $sl_2\mathbb{C}$. It says "If $V=\bigoplus V_{\alpha}$ and $W=\bigoplus W_{\beta}$ then ...
56 views
### complex eigenvectors with non zero real parts
I'm wondering about how to deal with complex numbers in eigenvectors that have non zero real parts, as in my eigenvector is $\bigl[\begin{smallmatrix}1-2i\\-1\end{smallmatrix}\bigr]$ that is supposed ...
81 views
179 views
### how do I prove that two matrices with same determinant and trace have different eigenvalues?
Assuming that they are both Hermetian, positive definite and have the same full rank. (To show the converse that if two matrices have the same eigenvalues, they must have the same determinant is ...
47 views
### Bounding the smallest eigenvalue of an ergodic Markov Chain
I am trying to prove that the smallest eigenvalue of an ergodic Markov chain is greater than -1. Can we do that using proof by contradiction, i.e. assuming the smallest eigenvalue being -1, etc.? The ...
101 views
### An eigenvector is a non-zero vector such that…
Various sources define eigenvalues and eigenvectors in slightly different ways (context independent). For example, both of the following definitions seem not to exclude the zero-vector as an ...
21 views
### Condition number and Chebyshev systems
Suppose I have a square matrix $A$ of size $n$ with elements $a_{mn}=\phi_m(x_n)$ where $\phi_m(x)$ can be thought of as a very friendly function: orthogonal, bandlimited, bounded and analytic. Also, ...
81 views
### Eigenvalues of $A$
Let $A$ be a $3*3$ matrix with real entries. If $A$ commutes with all $3*3$ matrices with real entries, then how many number of distinct real eigenvalues exist for $A$? please give some hint. ...
28 views
### Property of sequence of eigenvalues of an operator.
For a positive (self adjoint) operator $A$ with eigenvalues $\lambda_k$, is it possible to have the case when neither $\lambda_k\to \infty$ or $sup_k \lambda_k<\infty$ for example if a subsequence ...
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173 views
### Evaluating eigenvalues of a product of two positive definite matrices
Let $A,B\in M_n(\mathbb{R})$ be two symmetric positive definite matrices, i.e.: $$\forall x\in\mathbb{R}^n, x\neq 0, (Ax,x)>0, (Bx,x)>0,$$ where $(\cdot,\cdot)$ is the usual scalar product in ...
115 views
### A basic example of basis for eigenspaces
Hi I'm just learning how to find bases for eigenspaces and I ran into a very basic case that confused me. So the matrix is $$A = \begin{pmatrix}2 & 1\\0 & 2\end{pmatrix}$$ and $\lambda = 2$. ...
Assume I have some constant matrix $A$ to which I add a perturbation, resulting in $M(\epsilon )=A+\epsilon B$ the perturbed matrix ($B$ is constant as well), and that I can easily find the ...
### How to compute $\text{trace}((A+D)^{-1}A)$
Give a diagonal perturbation matrix $D$ (which is not an identity matrix), is there a simple way to compute $$\text{trace}((A+D)^{-1}A)$$ Or is there a good approximation?
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# Export the real roots of an equation
I wish to find the roots of y (with only its real part) for different values of r in range 1(0.01)30. I have tried the following:
b = 2.6666;
Pr = 10;
y = Roots[x^3 + x^2 (1 + b + Pr) + x (b + b (r - 1) + Pr*b) + 2*Pr*b (r - 1) == 0, x];
q1 = y /. {r -> {0}}
q2 = y /. {r -> {0.01}}
q3 = y /. {r -> {0.02}}
q4 = y /. {r -> {0.03}}
Is there a neater way to do this? Also, I want to export these roots to an excel sheet as shown here:
\$Version
"13.2.1 for Mac OS X ARM (64-bit) (January 27, 2023)"
Clear["Global*"]
b = 2.6666;
Pr = 10;
EDIT 2: Modified to return real part of roots rather than real roots.
(table = Prepend[
Table[{r, Sequence @@
Re[SolveValues[
x^3 + x^2 (1 + b + Pr) + x (b + b (r - 1) + Pr*b) +
2*Pr*b (r - 1) == 0, x]]},
{r, 0, 0.03, 0.01}],
{"r", "x1", "x2", "x3"}]) // Grid
EDIT: Or, for earlier versions without SolveValues, use
(table =
Prepend[Table[{r,
Sequence @@ Re[(x /.
Solve[x^3 + x^2 (1 + b + Pr) + x (b + b (r - 1) + Pr*b) +
2*Pr*b (r - 1) == 0, x])]}, {r, 0, 0.03,
0.01}], {"r", "x1", "x2", "x3"}]) // Grid
Export["test.csv", table];
Import["test.csv"] // Grid
• The posted code isn't getting solved for x. I am using Mathematica version 11. @Bob Hanlon May 29, 2023 at 5:24
• Could you please tell me how can I use it on version 11? @BobHanlon May 29, 2023 at 5:48
• Something is not right. For example, in range r=20(0.01)30, only a single root is displayed. I want to get all the three roots but with their real parts. @BobHanlon May 29, 2023 at 17:34
• Edited. Your range specification makes no sense in Mathematica. Presumably, you mean for the iterator to be {r, 20, 30, 0.01}` May 29, 2023 at 17:46
• Oh Yes I want the iterator to be {r, 20, 30, 0.01}. However, after using the edited code, there is still no difference in the output . @BobHanlon May 29, 2023 at 17:54
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A326604 G.f. A(x) satisfies x / Series_Reversion(x*A(x)) = (2*A(x) + 1+x)/3. 1
1, 1, 3, 21, 231, 3333, 58167, 1175877, 26827623, 679078677, 18844334727, 568229240901, 18491559492999, 645850960844469, 24099045218945031, 956889503377128261, 40291822946545245927, 1793614919867776690389, 84177429562216608349959, 4154548653801498090246597, 215137302566817565660007367, 11664210072689092804458508533 (list; graph; refs; listen; history; text; internal format)
OFFSET 0,3 COMMENTS a(k) = 6 (mod 9) when k = A191107(n) for n > 1 (conjecture). a(n) = 3*A249933(n) for n > 1. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..400 FORMULA G.f. A(x) satisfies: (1) A( 3*x/(2*A(x) + 1+x) ) = (2*A(x) + 1+x)/3. (2) A(x) = (1 + 2*A(x*A(x))) / (3-x). (3) A(x) = 1 + Sum_{n>=1} G^n(x) * 2^(n-1)/3^n where G(x) = x*A(x) and G^n(x) = G^{n-1}(x*A(x)) denotes iteration with G^0(x) = x. EXAMPLE G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 231*x^4 + 3333*x^5 + 58167*x^6 + 1175877*x^7 + 26827623*x^8 + 679078677*x^9 + 18844334727*x^10 +... such that x/Series_Reversion(x*A(x)) = (2*A(x) + 1+x)/3 = 1 + x + 2*x^2 + 14*x^3 + 154*x^4 + 2222*x^5 + 38778*x^6 + 783918*x^7 + 17885082*x^8 + 452719118*x^9 + ... ITERATIONS OF x*A(x). Let G(x) = x*A(x), then A(x) = 1 + G(x)/3 + G(G(x))*2/3^2 + G(G(G(x)))*2^2/3^3 + G(G(G(G(x))))*2^3/3^4 + G(G(G(G(G(x)))))*2^4/3^5 +... The table of coefficients in the iterations of x*A(x) begin: [1, 1, 3, 21, 231, 3333, 58167, 1175877, 26827623, ...]; [1, 2, 8, 58, 630, 8958, 154530, 3096330, 70161318, ...]; [1, 3, 15, 117, 1285, 18167, 310735, 6177745, 139076385, ...]; [1, 4, 24, 204, 2308, 32800, 559124, 11053668, 247451528, ...]; [1, 5, 35, 325, 3835, 55365, 946623, 18671961, 416326935, ...]; [1, 6, 48, 486, 6026, 89158, 1539350, 30423134, 677231222, ...]; [1, 7, 63, 693, 9065, 138383, 2427943, 48304893, 1076756889, ...]; [1, 8, 80, 952, 13160, 208272, 3733608, 75127944, 1682704256, ...]; [1, 9, 99, 1269, 18543, 305205, 5614887, 114768093, 2592154167, ...]; ... in which the following sum along column k equals a(k+1): a(2) = 3 = 1/3 + 2*2/9 + 3*4/27 + 4*8/81 + 5*16/243 + 6*32/729 +... a(3) = 21 = 3/3 + 8*2/9 + 15*4/27 + 24*8/81 + 35*16/243 + 48*32/729 + ... a(4) = 231 = 21/3 + 58*2/9 + 117*4/27 + 204*8/81 + 325*16/243 + 486*32/729 +... a(5) = 3333 = 231*2/3 + 630*2/9 + 1285*4/27 + 2308*8/81 + 3835*16/243 + 6026*32/729 +... MATHEMATICA nmax = 21; sol = {a[0] -> 1}; Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + 2 A[x A[x] + O[x]^(n+1)])/(3-x), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}]; sol /. Rule -> Set; a /@ Range[0, nmax] (* Jean-François Alcover, Nov 03 2019 *) PROG (PARI) /* Prints N terms using x/Series_Reversion(x*A(x)) = (2*A(x) + 1+x)/3 */ N = 30; {A=[1, 1]; for(i=1, N, A = concat(A, -3*Vec(x/serreverse(x*Ser(concat(A, 0))))[#A+1]); print1(i, ", ") ); A} CROSSREFS Cf. A120956, A249933. Sequence in context: A332708 A097329 A119097 * A008545 A005373 A078586 Adjacent sequences: A326601 A326602 A326603 * A326605 A326606 A326607 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 21 2019 STATUS approved
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Cody
Problem 2510. Solving Quadratic Equations (Version 1)
Solution 495494
Submitted on 3 Sep 2014 by Marisa
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
Test Suite
Test Status Code Input and Output
1 Pass
%% qe_correct1_1 = -1; qe_correct1_2 = -2; [qe_result1_1, qe_result1_2] = SolveQuadraticEquation(1, 3, 2); assert( (abs(qe_result1_1 - qe_correct1_1) < 0.0001 && ... abs(qe_result1_2 - qe_correct1_2) < 0.0001) || ... (abs(qe_result1_1 - qe_correct1_2) < 0.0001 && ... abs(qe_result1_2 - qe_correct1_1) < 0.0001) );
2 Pass
%% qe_correct2_1 = 0.224745; qe_correct2_2 = -2.22474; [qe_result2_1, qe_result2_2] = SolveQuadraticEquation(2, 4, -1); assert( (abs(qe_result2_1 - qe_correct2_1) < 0.0001 && ... abs(qe_result2_2 - qe_correct2_2) < 0.0001) || ... (abs(qe_result2_1 - qe_correct2_2) < 0.0001 && ... abs(qe_result2_2 - qe_correct2_1) < 0.0001) );
3 Pass
%% qe_correct3_1 = -1; qe_correct3_2 = -1; [qe_result3_1, qe_result3_2] = SolveQuadraticEquation(2, 4, 2); assert( (abs(qe_result3_1 - qe_correct3_1) < 0.0001 && ... abs(qe_result3_2 - qe_correct3_2) < 0.0001) || ... (abs(qe_result3_1 - qe_correct3_2) < 0.0001 && ... abs(qe_result3_2 - qe_correct3_1) < 0.0001) );
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https://math.stackexchange.com/questions/305002/finding-second-order-linear-homogenous-ode-from-the-fundamental-set-of-solutions
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# Finding second order linear homogenous ODE from the fundamental set of solutions
Find second order linear homogeneous ODE with constant coefficients if its fundamental set of solutions is {$e^{3t},te^{3t}$}.
Attempt: Had this question in my midterm. So, since the fundamental set of solutions is $$y=y_1+y_2=c_1e^{3t}+c_2te^{3t}$$ the characteristic equation of the second order ODE has only one root. I don't know what to do next. Help please.
• $(m-3)^2=0$.${}$ Feb 15, 2013 at 19:14
According to the theory of ODE with constant coefficients, the number of elements of a fundamental set of family of solutions coincides with the order of the ODE. So, your ODE here is second order. Moreover, if we consider the ODE with its standard form as $$ay''+by'+cy=0$$, so its associated auxiliary equation is $$am^2+bm+c=0.$$
Here you have $$m=3$$ of twice order, so the quadratic equation above is a complete square like $$(m-3)^2=0$$ and we know that in these cases we can find another solution, for example, by using the reduction method. This latter method gave us $$t\exp(3t)$$. Hence: $$(m-3)^2\longrightarrow m^2-6m+9=0\rightarrow y''-6y'+9y=0$$ is the equation.
• very nice solution and probably the one their prof was looking for, +1. Regards Feb 15, 2013 at 19:44
• Nice, Babak! good to see you 'round! +1 Feb 16, 2013 at 0:08
The eigenvalues and eigenvectors for the coefficient matrix $A$ in the linear homogeneous system:
$Y'= AY$ are $\lambda_{1} = 3$ with $v_1 =< a; b >$ and $\lambda_2 = 3$ with $v2 =< c; d >$
The fundamental form of the solution is:
$$Y = c_1 e^{3t}v_1 + c_2t e^{3t}v_2$$
Take the second derivative, $Y''$ for the DEQ.
Your original system will be of the form:
$$y'' - 6y' + 9 = 0$$
to give you the double eigenvalue $\lambda_{1,2} = 3$. You can actually solve this to find the corresponding eigenvectors.
Regards
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https://astronomy.com/magazine/ask-astro/2016/12/what-is-a-square-degree
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Tonight's Sky
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Searching...
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In many articles in your magazine, you refer to regions of the sky as being measured in “square degrees.” How exactly is a square degree defined, and how many of them does the sky contain?
James C. Reeder, Mililani Town, Hawaii
Pixabay/ Unsplash
Those studying the sky calculate square degrees the same way someone on Earth figures out the area of a rectangle or a circle. If a sky object measures 1° by 2°, it’s like a box measuring 1 foot by 2 feet. The areas are 2 square degrees and 2 square feet, respectively. And while you can designate the latter 2 ft2, there’s no easy abbreviation for square degrees.
We don’t always use degrees, however. For smaller objects, we use arcminutes or even arcseconds. Here’s an example: The Full Moon has an average diameter of 31.075 arcminutes, designated 31.075'. To find its area, use the formula for the area of a circle, πr2, where r is the radius. The answer, then, is π(15.5375)2, or approximately 758 square arcminutes.
That’s a valuable number to us because we often compare the size of a deep-sky object to that of the Full Moon. For example, if Galaxy X measures 19' by 11', it has an area of 209 square arcminutes. Divide that by 758, and you can say Galaxy X covers 28 percent as much area of sky as the Full Moon.
Finally, to find the number of square degrees in the entire sky, use the formula for the area of a sphere, 4πr2, where r = 1 radian (57.29577951°). So, the total area of the celestial sphere is 41,252.96125 square degrees. We have used this number occasionally to list the “percentage of sky” a certain constellation covers: Ursa Minor contains 255.86 square degrees. Therefore, it covers 0.62 percent of the entire sky.
Michael E. Bakich
Senior Editor
0
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Explore BrainMass
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# algebra - Linear Equation
1. Jackie buys a computer for \$3600. For tax purposes, she declares a linear depreciation (loss of value) of \$600 per year. Let y be the declared value of the computer after x years. If the linear relation of the depreciation model in this situation is given by 3600 - y = 600x
a. What is the slope of the line that models this depreciation?
b. Write a linear equation in slope-interception form to model the above depreciation equation.
c. Find the value of Jackie¿s computer after 4.5 years.
d. How many years will it take Jackie to fully depreciate her computer?
2. What kind of slope you would get if your employer decided to pay you the same salary whether you work overtime or not? Define the dependent and independent variables in this situation.
#### Solution Preview
(1) The linear equation of the depreciation model is y = -600x + 3600
(a) Slope = -600
(b) The slope-intercept form is y = -600x + 3600
(c) Plug in ...
#### Solution Summary
Neat and step-by-step solutions are provided.
\$2.19
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General
# How Much Is 29 Inches In Feet
If you are wondering how much is 29 inches in feet, you’re not alone. There are a number of people who are curious as well. This article will explain how to convert 29 inches to feet. Using the following table, you can find out how many feet 29 inches is. Once you know the measurement, you can then work out how many meters 29 inches is. This will save you a lot of time and effort.
Inches and feet are similar in length. They are a fraction of a foot. The foot is equal to 0.3048 meters or 2.42 feet. This measurement is commonly used in calculating lengths and is an important tool to understand measurements. A foot is the most commonly used customary unit of length in the United States. It is derived from the human leg and is subdivided into 12 inches.
In general, the length of a foot is one foot. The inch is a common unit of length and is divided into twelve inches. If you’re curious about the exact measurement of 29 inches, use the calculator below to find out the value in feet. If you’d like to see how much 29 inches is in feet, you can use a visual chart, which is easy to navigate and can be displayed in any screen resolution.
Another method to help you understand how much 29 inches is in feet is to create a chart. A visual chart is a great way to see how inches and feet relate. It is convenient and easy to use and can be displayed on a computer screen of any resolution. These charts show the relative values of inches and feet in different colors and lengths. If you’re looking to learn how to measure yourself or your kids, you can use a visual chart.
A visual chart is a great way to visualize the relationship between inches and feet. It will allow you to understand how much 29 inches in feet are in different ways. It will help you to better grasp the relation between the two measurements and make the most of your measurement. A visual chart will be helpful for you to visualize how 29 inches can be translated into feet. It will show the relationship between the two units easily. The resulting measurement will give you an idea of how long it really is.
You can also find a visual chart that will help you to understand the relationship between inches and feet. It will show the relative values of the two units. By using a visual chart, you can easily compare the different lengths of different measurements. You can even change the values of feet and inches. You can choose which of them works best for you. It is a quick and convenient way to learn about the relationship between feet and height.
Using a visual chart will also help you to understand the relationship between inches and feet. This chart will display the relative values of the two measurements. It will display the value of the inch in different color and length. You can also use this chart to understand the difference between inches and feet. A visual chart can make comparing the two units easier and more useful. There are several ways to calculate how much is 29 inches in feet.
Besides visual charts, you can also use a simple online calculator. These calculators will tell you how many feet are in 29 inches. Moreover, you can also get the value of the inches by converting them to feet. By using this chart, you will know how long are 29 inches in feet. A few other units will give you the same amount of feet as 29 inches. When you are converting from one length to another, you will get the value in seconds.
The answer to the question “how much is 29 inches in feet?” is not difficult to find. The conversion chart is made up of two different lengths and the ratio of an inch to a foot. If you want to convert inches to feet, simply enter the length in the first field and the width in the second. The result will be shown in feet in a matter of seconds. The converter will also show you the ratio between inches and feet.
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Kepler's laws
Kepler's second law
16Kp3 Kepler
Kepler's second law.
We now proceed to directly address Kepler's second law, the one which states that a ray from the Sun to a planet sweeps out equal areas in equal times. This ray is simply the vector r that we've been using. (And we shall continue to use it; r, remember, is defined as the vector from the Sun to the planet.)
What we're looking for is the area that this vector sweeps out. Imagine the planet at some time t = 0, and then imagine at a short time afterward t = deltat. In that time, the vector has moved by a short displacement
deltar = r|(t = deltat) - r|(t = 0). equation 1
The three vectors r|(t = 0), deltar, and r|(t = deltat) form a triangle. The area of this triangle closely approximates the area swept out by the vector r during that short time deltat.
We can write this small area represented by this triangle, deltaA, as one-half of the parallelogram defined by the vectors r and deltar, or
deltaA = (1/2) |r cross deltar|. equation 2
We'll divide both sides of this equation by deltat, the short time involved. Because of this, and the associative properties of the cross product, we find:
deltaA/deltat = (1/2) (1/deltat) |r cross deltar| equation 3
deltaA/deltat = (1/2) |(r cross deltar)/deltat| equation 4
deltaA/deltat = (1/2) |r cross (deltar/deltat)|. equation 5
As we choose smaller and smaller values of deltat, we get better and better approximations of the area swept out by the ray. If we let deltat approach zero by taking the limit of both sides, the approximation approaches the real value and we find that
dA/dt = (1/2) |r cross dr/dt| equation 6
or
dA/dt = (1/2) |r cross v|. equation 7
Knowing
l = m |r cross v| equation 8
and dividing both sides by m, we find
l/m = |r cross v|. equation 9
We can substitute this into our expression for dA/dt and find that
dA/dt = l/(2 m). equation 10
That is, the instantaneous time rate of change of area is the magnitude of the angular momentum divided by twice the mass of the planet. But we know that the mass of the planet is constant, and we also know from our work earlier that the angular momentum vector is constant (and thus its magnitude certainly is). Therefore, the time derivative of area swept out by this ray is constant. In other words, no matter where on the orbit the planet is, its ray still sweeps out the same amount of area. This is Kepler's second law.
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# Math experts
This Math experts provides step-by-step instructions for solving all math problems. We will also look at some example problems and how to approach them.
## The Best Math experts
In this blog post, we will show you how to work with Math experts. There are many ways to solve a right triangle, but one of the most common methods is using the Pythagorean Theorem. . This theorem can be used to find the missing sides of a right triangle if the length of the hypotenuse and one other side are known.
Log equations can be solved by isolating the log term on one side of the equation and using algebra to solve for the unknown. For example, to solve for x in the equationlog(x) = 2, one would isolate the log term on the left side by subtracting 2 from each side, giving the equation log(x) - 2 = 0. Then, one can use the fact that the log of a number is equal to the exponent of that number to rewrite the equation
Basic math is the foundation of all mathematics. It includes the four operations of addition, subtraction, multiplication, and division. It also includes the basic concepts of fractions, decimals, and percentages. Basic math is essential for everyday life and for more advanced mathematics.
I'm not a math person, but I have a few tricks up my sleeve when it comes to solving math problems. First, I always start by reading the problem and trying to understand what it's asking. Once I have a good understanding of the problem, I'll start by solving the easy parts first. If there are any parts of the problem that are giving me trouble, I'll take a step back and try to simplify it. And if all else fails, I'll ask a
Math can be a difficult subject for many college students. Thankfully, there are now many online resources that can provide assistance with math problems. These websites typically provide step-by-step solutions to problems, as well as explanations of the theory behind the problems. This can be a great help for students who are struggling with math, and it can also be a good way for students to check their work.
There are a few steps that need to be followed when solving limits. First, it is important to understand what the limit is and what it is trying to find. Once this is understood, the next step is to identify what type of limit it is. There are three main types of limits: one-sided, two-sided, and infinity limits. After the type of limit is identified, the appropriate formula can be used to solve for the limit.
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# Mirror, Mirror in the Box
Merlin the magician has a magic box which plays tricks with light. He places mirrors in the box so that when a light beam enters, it bounces around before exiting:
He shows you the box, and lets you fire light beams into it, but he does not let you see how the mirrors are set up. Is it always possible for you to figure out where all the mirrors are inside the box?
### Extensions:
• If you were to randomly position the pairs of coloured arrows above, is it always possible for a set of mirrors to be placed so that a light beam starting at a specific colour emerges from the box at the same colour?
• Is it possible to place at least one mirror so that all light beams pass straight through the box as if there were no mirrors inside?
• Imagine an infinitely large box with the following pattern of mirrors:
• How many different shapes can a light beam trace in this infinite box? The red dot is one position where a light beam could start. You can place dots of different colours to help you.
• A light beam starts on the red intersection. After a millisecond it is discovered that a light beam is 5 units higher. How many units left or right may it be?
• A light beam starts on the red intersection. After a millisecond it is discovered that a light beam is 5 units higher. How many units left or right may it be?
### The Math in This Problem:
This math puzzle challenges students to work with light beams reflected off mirrors within various boxes. To simplify the notion of reflection, which is a fundamental notion of mathematics, this brainteaser introduces it using 90 degree angles.
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https://www.physicsforums.com/threads/block-on-wedge-which-is-not-fixed-find-minimum-m-to-keep-the-wedge-at-rest.773055/
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# Homework Help: Block on wedge which is not fixed.Find minimum μ to keep the wedge at rest.
1. Sep 26, 2014
### Satvik Pandey
1. The problem statement, all variables and given/known data
Consider a wedge of mass m whose upper surface is a quarter circle and is smooth. The lower part is rough. A small block of same mass is kept at the topmost position and the initial position is shown in figure. The block slides down the wedge.
Find the minimum value of coefficient of friction between floor and lower surface of wedge so that the wedge remains stationery during the block's motion on it.
2. Relevant equations
3. The attempt at a solution
I tried to find the angle 'theta' at which horizontal force on the wedge is maximum.
By conservation of energy
$mgr=mgr(1-sin\theta )+\frac { m{ v }^{ 2 } }{ 2 }$
So ${ v }^{ 2 }=2grsin\theta$
Horizontal force on the wedge(Fx) is $\frac { m{ v }^{ 2 } }{ r } cos\theta$
So $Fx=\frac { m }{ r } 2grsin\theta cos\theta$
$Fx=mgsin2\theta$
For Fx to be maximum $d(Fx)/d\theta=0$
so
$mg\frac { d }{ d\theta } sin2\theta =0$
$\theta$=45
So ${ v }^{ 2 }=\sqrt { 2 } \ gr$
Normal force on the wedge = $mg+mg+\frac { m{ v }^{ 2 } }{ 2 } sin\theta$=3mg
Horizontal force on wedge= $mgsin2\theta$=mg
As wedge should remain at rest so
$3\mu mg=mg$
So $\mu =1/3$
But this is not correct.Where did I go wrong?
2. Sep 26, 2014
### ehild
Are you sure? The force of friction depends on the normal force between the ground and the wedge, and it is not equal to the weight of the wedge.
The horizontal force of the wedge is the horizontal component of the normal force between the small block and the wedge. The normal force is not equal to the centripetal force. Do not forget agravity.
ehild
3. Sep 26, 2014
### Satvik Pandey
From figure 2
$N=\frac { m{ v }^{ 2 } }{ R } +mgcos(90-\theta )$
or $N=\frac { m{ v }^{ 2 } }{ R } +mgsin\theta$
Horizontal force in wedge $Fx=Ncos\theta =\left( \frac { m{ v }^{ 2 } }{ R } +mgsin\theta \right) cos\theta$
From figure 3
$R=mg+Nsin\theta$
$R=mg+\left( \frac { m{ v }^{ 2 } }{ R } +mgsin\theta \right) sin\theta$
Are these equations correct?
Is my expression of v2 in #post1 correct?
4. Sep 26, 2014
### ehild
And you got already that $v^2=2gR\sin(\theta)$, so N= ??
Horizontal force in wedge $Fx=Ncos\theta =\left( \frac { m{ v }^{ 2 } }{ R } +mgsin\theta \right) cos\theta$
Certainly not, R is the radius, a length, it can not be force ...
5. Sep 26, 2014
### Satvik Pandey
$N=\frac { m2gR\sin (\theta ) }{ R } +mgsin\theta =3 mg\sin (\theta )$
$Fx=3 mg\sin (\theta )cos\theta$......(1)
The 'R' in the RHS is not radius.It is reactionary force on wedge.Sorry I should have choose other variable for it, let it be A.
So
$A=mg+Nsin\theta$
$A=mg+\left( \frac { m{ v }^{ 2 } }{ R } +mgsin\theta \right) sin\theta$
Please consider reactionary force on wedge in figure(3) in #post 3 be A.
6. Sep 26, 2014
### Satvik Pandey
On differentiating eq(1) wrt to $\theta$ and equating it to 0 I got $\theta=45$ maximum value of $Fx=3mg/2$
7. Sep 26, 2014
### ehild
You need maximum coefficient of friction, not the maximum horizontal force. The force of friction also depends on the normal force and on theta.
And substitute for v2 also in the expression of A.
8. Sep 26, 2014
### Satvik Pandey
$A=mg+3mg{ sin }^{ 2 }\theta$
I thought by equating $\mu A=Fx$ I can solve for $\mu$???
9. Sep 26, 2014
### ehild
Solve for mu and find the smallest value.
10. Sep 26, 2014
### Satvik Pandey
Should $\theta=45$?
11. Sep 26, 2014
### ehild
No, why? Solve for mu in terms of theta, and find the minimum value of mu.
12. Sep 26, 2014
### Satvik Pandey
I got $\mu=3/5$
Oh! Sorry.
Putting value in $\mu A=Fx$
I got $μ(mg+3mg{ sin }^{ 2 }\theta )=3mgsin\theta cos\theta$
or $μ=\frac { 3sin\theta cos\theta }{ 1+3{ sin }^{ 2 }\theta }$
In order to have $\mu$ maximum
$\frac { d }{ d\theta } \left( \frac { 3sin\theta cos\theta }{ 1+3{ sin }^{ 2 }\theta } \right) =0$
or $\frac { d }{ d\theta } \left( \frac { sin\theta cos\theta }{ 1+3{ sin }^{ 2 }\theta } \right) =0$
or $\frac { (1+3{ sin }^{ 2 }\theta )\frac { d }{ d\theta } (sin\theta cos\theta )-(sin\theta cos\theta )\frac { d }{ d\theta } (1+3{ sin }^{ 2 }\theta ) }{ { (1+3{ sin }^{ 2 }\theta ) }^{ 2 } } =0$
or $(1+3{ sin }^{ 2 }\theta )\frac { d }{ d\theta } (sin\theta cos\theta )-(sin\theta cos\theta )\frac { d }{ d\theta } (1+3{ sin }^{ 2 }\theta )=0$
or $(1+3{ sin }^{ 2 }\theta )cos2\theta -(sin\theta cos\theta )(3sin2\theta )=0$
or $2(1+3{ sin }^{ 2 }\theta )cos2\theta =2(sin\theta cos\theta )(3sin2\theta )$
or $2(1+3{ sin }^{ 2 }\theta )cos2\theta =3{ sin }^{ 2 }2\theta$
or $2(1+3{ sin }^{ 2 }\theta )(1-2{ sin }^{ 2 }\theta )=3{ sin }^{ 2 }2\theta$
How to solve for $\theta$?
13. Sep 26, 2014
### ehild
$\mu$ need to be minimum.
Change to double angles. You have turned $\sin(\theta)\cos(\theta)$ to $0.5 \sin(2\theta)$ already, but remember the very important formulas
$\cos^2(\theta)= \frac{1+\cos(2\theta )}{2}$
and
$\sin^2(\theta)= \frac{1-\cos(2\theta) }{2}$
ehild
14. Sep 27, 2014
### Satvik Pandey
That was a typo.
Is $\theta ={ cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \times \frac { 1 }{ 2 }$??
15. Sep 27, 2014
### Satvik Pandey
So $\theta =26.565$
As $μ=\frac { 3sin\theta cos\theta }{ 1+3{ sin }^{ 2 }\theta }$
or $μ=\frac { 3sin2\theta }{ 2(1+3{ sin }^{ 2 }\theta ) } =\frac { 3\times 0.8 }{ 3.199 } =0.75$
$\mu=0.75$
Thank you ehild.
16. Sep 27, 2014
### ehild
Splendid !
ehild
17. Sep 27, 2014
### Satvik Pandey
Thank you ehild. I could't have solved it without your help.:)
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Circles & Parabola (Posted on 2013-07-13)
Two circles with equal radii are externally tangent
at a point on the parabola y = x2. One of the circles
is also tangent to the x-axis while the other is also
tangent to the y-axis. Find the radius of both circles.
Submitted by Bractals No Rating Solution: (Hide) Let (a,r) and (r,b) be the centers of the circles tangent to the x-axis and y-axis respectively (where r is the radius desired). Let (c,c2) be the point of tangency on the parabola (WOLOG the first quadrant branch will be used). The centers of the circles must lie on the normal line at the point (c,c2) (see my reply to Charlie's post for a correction to the problem):``` c2-r c2-b -1 ------ = ------ = ---- (1) c-a c-r 2c``` The distance from the centers to the point (c,c2) is r:``` (c2-r)2 + (c-a)2 = (c2-b)2 + (c-r)2 = r2 (2) ``` Combining (1) and (2) to eliminate the coordinates a and b to give:``` (c2-r)2 + 4c2(c2-r) = r2 (3) and (c-r)2/4c2 + (c-r)2 = r2 (4) ``` Which in turn give``` r = wc2/(w+1) (5) and r = wc/(w+2c) (6) where w2 = 4c2+1.``` Combine (5) and (6) to find c:``` wc2/(w+1) = wc/(w+2c) ⇒ 8c2-9c+2 = 0 ⇒ c = [9+s√(17)]/16 (7) where s = ±1.``` The point (c,c2) is the midpoint of the line segment joining the centers of the circles. Therefore,` c = (a+r)/2 (8)` Combining (1) and (8) gives``` c2-r 1 ------ = ---- (9) c-r 2c``` Therefore,``` c(2c2-1) r = ---------- (10) 2c-1``` Combining (7) and (10) gives``` r = 3[23-s√(17)]/128 (11) ``` QED
Comments: ( You must be logged in to post comments.)
Subject Author Date Solution Harry 2013-07-14 06:43:52 computer approximation Charlie 2013-07-13 17:07:42 a start Charlie 2013-07-13 15:46:14 re: thoughts - Problem Correction Bractals 2013-07-13 14:30:53 thoughts Charlie 2013-07-13 13:55:05
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# Thread: having difficulty with question similiar to previous post
1. ## having difficulty with question similiar to previous post
Still having difficulty solving for u and s
Question is
It is known that the IQ scores of workers in a certain company are normally distributed with a standard deviation of ten. If 0.13% of workers have IQs in excess of 130, calculate the mean IQ.
This is what I gather from question
s=10
p(x>130)=P(z>.13)
z value in chart for .13 is 1.13
z=x-u/s or -1.13=x-u/10
stuck at this point - I think I am having problem with rearranging formual to solve for u and s as example questions in course have only had me solve for x and z not u and s
2. You actually need P(z > 0.0013), which is Z=3.011
You're given s = 10 and x = 130
z = (x - u)/s
3.011 = (130 - u) / 10
Solve for u.
3. thanks got this one,
#### Posting Permissions
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https://www.teacherspayteachers.com/Product/3rd-Grade-Math-STAAR-Prep-14-SPANISH-No-Prep-Games-by-Marvel-Math-3185657
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# Main Categories
Total:
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# 3rd Grade Math STAAR Prep: 14 SPANISH No Prep Games by Marvel Math
Product Description
This is the SPANISH version of these games. The English version is available here.
This is a fun set of 14 low prep, STAAR test formatted games that cover the 3rd grade math readiness TEKS. STAAR prep can be complete drudgery, so I use these games in conjunction with released test items to maximize student engagement in preparation for the STAAR math test. I also know your time is valuable, so these are print-and-go games.
I created at least one game for each readiness TEK, and two games for the TEKS that had too much content for just one game. I studied the released STAAR items from the past 2 years and the TEKS to create games that covered essential content and resembled STAAR test questions. I couldn’t be more excited about our test prep!
One added benefit of games over worksheets is the flexibility to use them time and time again. The games are built so that different combinations of numbers and problem stems will arise each time you play. Even if problems are repeated they offer students an opportunity to build fluency.
I am using these games in conjunction with the Practice STAAR test I created. Using the student data tool in that test, I can easily see which Readiness standards students have not mastered and use these games prescriptively with them at my small group table.
3rd Grade Practice STAAR Math Test
I also use my FUN SPINNERS
with my games instead of paperclips!
*****************************************************************************
Included Games:
Compose Numbers 3.2A
Compare Numbers 3.2D
Equivalent Fractions with Fraction Bars 3.3F
Equivalent Fractions with Number Lines 3.3F
Compare Fractions 3.3H
Problem Solving: 1 & 2 Step Addition/Subtraction 3.4A
Problem Solving: 1 & 2 Step Multiplication/Division 3.4K
Represent 1 & 2 Step Addition/Subtraction 3.5A
Represent 1 & 2 Step Multiplication/Division 3.5B
Table Relationships 3.5E
Classify Figures 3.6A
Determine Area 3.6C
Calculate Perimeter 3.7B
Summarize Data 3.8A
*****************************************************************************
If your students need more skills practice, check out my \$1 Printables. Students solve a problem and then follow a maze to the next problem. They are a BIG HIT with my students!
3-Digit Subtraction with Regrouping Maze Printables
Subtracting Across Zeros to 1,000 Maze Printables
3-Digit Addition with Regrouping Maze Printables
2-by-1 Multiplication Maze Printables
2-by-2 Multiplication Maze Printables
Long Division Maze Printables
Here are some of my TEKS-aligned task cards and activities for 3rd grade:
3.7B Bake Shop Perimeter- Find the Missing Side
3.4A One and Two Step Problem Solving with Addition and Subtraction
3.2A Place Value- Compose and Decompose Numbers
3.8A 3rd Grade Graphing Booklet: Dot Plots, Pictographs, Bar Graphs
*****************************************************************************
How I have used this product in my class:
I modeled the games in a me vs. the class game, which was a perfect way to review each concept. Then I let my students play in pairs. I included answer keys for games that needed them, so that students could check one another’s work and monitor their learning. Students that were still struggling with each concept played the game at my table in pairs, where I could support them.
I have found that games make for a perfect small group lesson. “Come to my table to play a math game with me,” is so much better than, “Come to my table to work on multi-step problems.” Games are enticing and engaging for our students, especially those who struggle in math. I also have the opportunity to model my thinking and math strategies out loud for my group when it is “my turn” in the game. I can adjust my strategies depending on the level of my math group and add on strategies as we play the game, since I get to use my turns as mini-lessons. The best part is….my students don’t even realize this!!! During my group’s turn, they work out the problem individually on a whiteboard, so I can monitor their individual progress.
I am also planning to use these games after the math STAAR test to keep the content fresh for my students. They will be great games to use the last 2 weeks of school as classes are winding down.
I hope you and your class have as much fun with these games as we did!
-Melissa Johnson at Marvel Math
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×
Get Full Access to Basic Engineering Circuit Analysis - 11 Edition - Chapter 2 - Problem 2.69
Get Full Access to Basic Engineering Circuit Analysis - 11 Edition - Chapter 2 - Problem 2.69
×
Determine the total resistance, RT, in the circuit in Fig.
ISBN: 9781118539293 159
Solution for problem 2.69 Chapter 2
Basic Engineering Circuit Analysis | 11th Edition
• Textbook Solutions
• 2901 Step-by-step solutions solved by professors and subject experts
• Get 24/7 help from StudySoup virtual teaching assistants
Basic Engineering Circuit Analysis | 11th Edition
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5
Problem 2.69
Determine the total resistance, RT, in the circuit in Fig. P2.69.
Step-by-Step Solution:
Step 1 of 3
ENGE 201 Notes Hexadecimal numbers Base 16 number system; used as a shorthand of binary Values range from 0-F o 1, 2, 3, ,4 5, 6, 7, 8, 9, A, B, C, D, E, F o positional multipliers are powers of 16: 16^1, 16^2, etc. Hex Decimal Binary Hex cont. Decimal Binary 0 0 0000 8 8 1000 1 1 0001 9 9 1001 2 2 0010 A 10 1010 3 3 0011 B 11 1011 4 4 0100 C 12 1100 5 5 0101 D 13 1101 6 6 0110 E 1
Step 2 of 3
Step 3 of 3
ISBN: 9781118539293
Since the solution to 2.69 from 2 chapter was answered, more than 481 students have viewed the full step-by-step answer. This textbook survival guide was created for the textbook: Basic Engineering Circuit Analysis, edition: 11. Basic Engineering Circuit Analysis was written by and is associated to the ISBN: 9781118539293. The full step-by-step solution to problem: 2.69 from chapter: 2 was answered by , our top Engineering and Tech solution expert on 11/23/17, 05:00AM. This full solution covers the following key subjects: Circuit, determine, fig, resistance. This expansive textbook survival guide covers 15 chapters, and 1430 solutions. The answer to “Determine the total resistance, RT, in the circuit in Fig. P2.69.” is broken down into a number of easy to follow steps, and 11 words.
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# HELP! Sunflower Seed Mixture Math Problem: Agway Gardens has sunflower seeds that sell for \$0.49 per pound?
Agway Gardens has sunflower seeds that sell for \$0.49 per pound and gourmet bird seed mix that sells for \$0.89 per pound. How much of each must be mixed to get a 20-pound mixture that sells for \$0.73 per pound?
I came up with this equation:
let S = Sunflower seeds
B = Bird seed mix
I came up with this equation: (0.49S + 0.89B) / 20 = 0.73
But I couldn't find the solution.
as you said
s= sunflower seed in pounds
b= bird seed mix in pounds
total weight = 20 lb
then , s= 20-b
(0.49s+0.89b) =20*0.73
0.49(20-b)+0.89b=20*0.73
0.49*20-0.49b+0.89b=20*0.73
9.8+0.4b=14.6
0.4b=14.6-9.8=4.8
b=4.8/0.4=12
so bird seed mix=12 pounds
sun flower seeds=20-12=8 pounds
check
0.49*8+0.89*12=3.92+10.68=14.6
20*0.73=14.6
#1
You need to be clearer in you definitions. "S" is not just sunflower seeds, it is the number of pounds of sunflower seeds to use
S + B = 20
Now you have two equations in two unknowns, and can solve.
#2
since there are two unknowns you will need two equations.
since S and B represent the total amount in pounds and you know you need 20 pounds, the 2nd equation would be
S + B = 20
then you can solve both equations noting that S = 20 - B so
(.49*(20-B+.89*B)/20=.73
solve for B then plug that into either of the first equations to solve for A
#3
If S is the number of pounds of sunflower seeds, 20-S is the number of pounds of bird seed in the mix. Then your equation becomes [0.49S + 0.89(20-S)]/20 = 0.73
Now you can solve for S, and hence for B.
#4
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# What is 24k in scientific notation?
In scientific notation, 24k would be written as 2.4 x 10^4. This representation simplifies large numbers by expressing them in the form of a coefficient multiplied by a power of 10. In the case of 24k, the coefficient is 2.4 and the exponent is 4, indicating that the number should be multiplied by 10 four times.
Scientific notation is especially useful for denoting very large or very small numbers in a concise and standardized format. By breaking down numbers into a coefficient and an exponent, scientific notation allows for easier calculation and comparison of values across different scales. Therefore, expressing 24k as 2.4 x 10^4 provides a clear and efficient way to convey its magnitude in the realm of scientific and mathematical contexts.
## Understanding Scientific Notation
Scientific notation is a valuable mathematical concept often used in various areas of science. It is incredibly crucial for dealing with very large or very small numbers, making the field of calculations far more manageable and comprehensible.
# 24k in Scientific Notation
24k is a commonly used abbreviation for 24,000. When expressed in scientific notation, 24k (24,000) becomes 2.4 x 104.
## Breaking Down Scientific Notation
Scientific notation follows a specific format: a number between 1 and 10 (known as the coefficient) multiplied by ten to the power of an exponent. In the case of 24,000 or 24k, this is broken down into the coefficient 2.4 and the exponent 4. Hence, 2.4 x 104.
### The Basis of Exponents in Scientific Notation
In scientific notation, the exponent dictates how many places the decimal point should move. A positive exponent indicates the direction towards the right or an increase by a factor of ten, whereas a negative exponent points to the left or represents division by a factor of ten. For 24k or 2.4 x 104, the exponent 4 indicates that the decimal point in 2.4 should move four places to the right to get 24000.
## Real World Application of Scientific Notation
Applying scientific notation to everyday life, 24k is often seen in the context of money where ‘k’ denotes thousands. So when we say 24k, it means \$24,000. This way, scientific notation simplifies large numbers and makes comparisons more manageable. Hence, scientific notation isn’t limited to classrooms or scientists but extends its benefits across different industries and professions.
### Scientific Notation in Mathematics and Beyond
Scientific notation transcends beyond just simplifying large numbers; it carries significance in various subjects, including physics and astronomy, where distances and measurements are seldom in manageable numbers. Furthermore, understanding scientific notation like 24k or 2.4 x 104 allows students and professionals to better comprehend and tackle complex mathematical and scientific challenges.
## The Rationale Behind Using the ‘k’ in 24k
‘K’ is a common notation representing ‘kilo-, ‘ which derives from the Greek word ‘chilioi,’ meaning thousand. In terms of finances, 24k undoubtedly stands for 24 thousand dollars. Thus the term 24k might be seen used commonly in various scenarios, particularly related to earnings, price mentioning, and more.
In scientific notation, 24k can be expressed as 2.4 x 10^4. This notation is a concise and standardized way to represent large numbers, making it easier to work with in mathematical equations and scientific calculations.
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# The electric field intensity produced by the radiation coming from
Question:
The electric field intensity produced by the radiation coming from a $100 \mathrm{~W}$ bulb at a distance of $3 \mathrm{~m}$ is $\mathrm{E}$. The electric field intensity produced by the radiation coming from $60 \mathrm{~W}$
at the same distance is $\sqrt{\frac{\mathrm{x}}{5} \mathrm{E} \text {. Where the value }}$
of $x=$ ___________
Solution:
(3)
$c \in_{0} E^{2}=\frac{100}{4 \pi \times 3^{2}}$
$c \in_{0}\left(\sqrt{\frac{x}{5}} E\right)^{2}=\frac{60}{4 \pi \times 3^{2}}$
$\Rightarrow \frac{x}{5}=\frac{3}{5}$
$\Rightarrow x=3$
Leave a comment
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# Strategies for dividing by tenths
Sal uses equivalent fractions to divide both whole numbers and decimals by tenths.
## Want to join the conversation?
• Does anyone else think that dividing with decimals confusing.
• There is another trick. You can multiply both numbers by 10, or 100 etc.
In zhe example above 0.8÷0.25, we can multiply all by 100 and get:
80÷25.
• So 1.2 is like 12,right? I understand it, but why is it like that? Because 1 is not 12 so that means 1.2 is not 12. Because you take away the decimal then it is 12.
• Because when you divide by a decimal, you remove move the decimal place to the left however many digits until it is a whole number. Then you divide the number and move the decimal to the right however many times you moved it to the left.
e.g. 20/0.5 20/5= 4.0 = 40
But when you divide a decimal by a decimal, you move the decimal to the left until it is a whole number on both numbers, then after you divide you move it to the right for the combined number of times you moved to the left.
Hope this helps :)
• Math is good for your brain
• Why / how is 4.2/0.3 the same as 42/3?
• It's a matter of moving the decimal points.
If you move the decimal point on 4.2 to the right it becomes 42, but you also have to do the same thing on the denominator so 0.3 becomes 3. Put them together and you have 42/3
If you're not sure, divide 4.2 by 0.3 and then divide 42 by 3. You'll find that you get the same answer.
• When dividing decimals, what is the reason for removing the decimals and making them whole numbers? If I were to divide a tenth by a hundredth (e.g., 0.2/0.15), would I multiply by 10/100? Can and should the decimals always be removed when dividing numbers that have decimals? What is the reason for removing the decimals?
• When multiplying the decimal moves the right because the numbers value increases. But when you divide your decimal goes to the left because your number decreases
• I need help can someone please help me? I don't get what I have to do. Can someone that knows how to do this help me?
• Yes, of course. So for example, you have 2.5 divided by 5. The first thing to do is to convert 2.5 to a whole number by multiplying by ten. But of course, what you do to one number you have to do to another. So 2.5 is now 25 and 5 is now 50. Divide them and you will get 2.
Any other questions?
(1 vote)
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# 133095 (number)
133,095 (one hundred thirty-three thousand ninety-five) is an odd six-digits composite number following 133094 and preceding 133096. In scientific notation, it is written as 1.33095 × 105. The sum of its digits is 21. It has a total of 4 prime factors and 16 positive divisors. There are 67,104 positive integers (up to 133095) that are relatively prime to 133095.
## Basic properties
• Is Prime? No
• Number parity Odd
• Number length 6
• Sum of Digits 21
• Digital Root 3
## Name
Short name 133 thousand 95 one hundred thirty-three thousand ninety-five
## Notation
Scientific notation 1.33095 × 105 133.095 × 103
## Prime Factorization of 133095
Prime Factorization 3 × 5 × 19 × 467
Composite number
Distinct Factors Total Factors Radical ω(n) 4 Total number of distinct prime factors Ω(n) 4 Total number of prime factors rad(n) 133095 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 133,095 is 3 × 5 × 19 × 467. Since it has a total of 4 prime factors, 133,095 is a composite number.
## Divisors of 133095
16 divisors
Even divisors 0 16 8 8
Total Divisors Sum of Divisors Aliquot Sum τ(n) 16 Total number of the positive divisors of n σ(n) 224640 Sum of all the positive divisors of n s(n) 91545 Sum of the proper positive divisors of n A(n) 14040 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 364.822 Returns the nth root of the product of n divisors H(n) 9.4797 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors
The number 133,095 can be divided by 16 positive divisors (out of which 0 are even, and 16 are odd). The sum of these divisors (counting 133,095) is 224,640, the average is 14,040.
## Other Arithmetic Functions (n = 133095)
1 φ(n) n
Euler Totient Carmichael Lambda Prime Pi φ(n) 67104 Total number of positive integers not greater than n that are coprime to n λ(n) 8388 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 12389 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 67,104 positive integers (less than 133,095) that are coprime with 133,095. And there are approximately 12,389 prime numbers less than or equal to 133,095.
## Divisibility of 133095
m n mod m 2 3 4 5 6 7 8 9 1 0 3 0 3 4 7 3
The number 133,095 is divisible by 3 and 5.
## Classification of 133095
• Arithmetic
• Deficient
• Polite
• Square Free
### Other numbers
• LucasCarmichael
## Base conversion (133095)
Base System Value
2 Binary 100000011111100111
3 Ternary 20202120110
4 Quaternary 200133213
5 Quinary 13224340
6 Senary 2504103
8 Octal 403747
10 Decimal 133095
12 Duodecimal 65033
20 Vigesimal gcef
36 Base36 2up3
## Basic calculations (n = 133095)
### Multiplication
n×y
n×2 266190 399285 532380 665475
### Division
n÷y
n÷2 66547.5 44365 33273.8 26619
### Exponentiation
ny
n2 17714279025 2357681966832375 313795681375554950625 41764636212679486153434375
### Nth Root
y√n
2√n 364.822 51.0568 19.1003 10.5884
## 133095 as geometric shapes
### Circle
Diameter 266190 836261 5.5651e+10
### Sphere
Volume 9.87584e+15 2.22604e+11 836261
### Square
Length = n
Perimeter 532380 1.77143e+10 188225
### Cube
Length = n
Surface area 1.06286e+11 2.35768e+15 230527
### Equilateral Triangle
Length = n
Perimeter 399285 7.67051e+09 115264
### Triangular Pyramid
Length = n
Surface area 3.0682e+10 2.77855e+14 108672
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# Reinforcement Schedules Match the type of reinforcement schedule
• Slides: 49
Reinforcement Schedules Match the type of reinforcement schedule to the example that best defines it.
Reinforcement Schedule Review RATIOS = RESPONSES • Fixed Ratio (FR) – Reinforcement occurs after a fixed number of responses – Results in a burst-pause-burst-pause pattern • Variable Ratio (VR) – Reinforcement occurs after an average number of responses. Number of responses required for reinforcement is unpredictable. – Results in high, steady rates of responding with almost no pauses
Reinforcement Schedule Review INTERVALS = TIME • Fixed Interval (FI) – A reinforcer is delivered for the first response after a preset time interval has elapsed. – Number of responses tends to increase as the time for the next reinforcer draws near then long pause occurs after reinforcement. • Variable Interval (VI) – A reinforcer is delivered for the first response after an average time interval has elapsed. The interval is unpredictable. – Number of responses will be moderately steady because reinforcement could occur at any time.
Example 1 ANSWER • FIXED RATIO (FR)
Example 2 Although farmers put out about the same amount of labor each year, sometimes they sell their grain at harvest time, sometimes in January, and sometimes in late March.
Example 2 ANSWER • VARIABLE INTERVAL (VI)
Example 3 You are fly fishing in a great trout stream. Sometimes you catch a fish after only 4 or 5 casts; sometimes it takes 10 or 20 casts.
Example 3 ANSWER • VARIABLE RATIO (VR)
Example 4 A bank employee is given a bonus of a \$30 restaurant coupon for every 4 hours of overtime he volunteers to work.
Example 4 ANSWER • FIXED INTERVAL (FI)
Example 5 A young child demands a candy bar each time she is in the checkout line at the grocery store. Sometimes the parent gives her one and sometime he doesn’t.
Example 5 ANSWER • VARIABLE RATIO (VR)
Example 6 A piano teacher gives a 6 -year-old student a sticker for each song he plays well.
Example 6 ANSWER • FIXED RATIO (FR)
Example 7 You are still fishing from a dock. You sit with your line in the water and wait for the fish to strike. Sometimes it takes only 20 minutes, and sometimes it takes an hour or more.
Example 7 ANSWER • VARIABLE INTERVAL (VI)
Example 8 You do piece work in a factory. For every 3 sleeves you sew on garments, you get \$0. 50.
Example 8 ANSWER • FIXED RATIO (FR)
Example 9 For each half day that a second grader pays attention and works well, the teacher gives her a “Good Behavior” dollar that she can trade in at the end of the week for desired objects from the store.
Example 9 ANSWER • FIXED INTERVAL (FI)
Example 10 A vacuum cleaner salesperson does numerous presentations in people’s homes and gets paid a bonus each time he or she sells a vacuum. Sometimes the salesperson does 5 presentations before a customer buys a vacuum, sometimes 2 customers in a row purchase the product.
Example 10 ANSWER • VARIABLE RATIO (VR)
Example 11 Your paycheck arrives on the 1 st and the 15 th of every month.
Example 11 ANSWER • FIXED INTERVAL (FI)
Example 12 Buying state lottery tickets & winning.
Example 12 ANSWER • Variable Ratio (VR)
Example 13 A hotel maid may take a 15 -minute break only after having cleaned three rooms.
Example 13 ANSWER • Fixed Ratio (FR)
Example 14 Watching and seeing shooting stars on a dark night.
Example 14 ANSWER • Variable Interval (VI)
Example 15 A teenager receives an allowance every Saturday.
Example 15 ANSWER • Fixed Interval (FI)
Example 16 Checking the front porch for a newspaper when the delivery person is extremely unpredictable.
Example 16 ANSWER • Variable Interval (VI)
Example 17 A professional baseball player gets a hit approximately every third time at bat.
Example 17 ANSWER • Variable Ratio (VR)
Example 18 Checking the oven to see if chocolate chip cookies are done, when the baking time is known.
Example 18 ANSWER • Fixed Interval (FI)
Example 19 A blueberry picker receives \$1 after filling 3 pint boxes.
Example 19 ANSWER • Fixed Ratio (FR)
Example 20 A charitable organization makes an average of ten phone calls for every donation it receives.
Example 20 ANSWER • Variable Ratio (VR)
Example 21 Calling a garage mechanic to see if your car is fixed yet.
Example 21 ANSWER • Variable Interval (VI)
Example 22 A student’s final grade improves one level for every three book reviews submitted.
Example 22 ANSWER • Fixed Ratio (FR)
Example 23 Going to the cafeteria to see if the next meal is available.
Example 23 ANSWER • Fixed Interval (FI)
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dgenchev
# Thinking About Currency Risk: To Hedge or Not to Hedge
In my previous article I advanced the idea of investing globally and went over the major concerns that one faces in doing so. If you are inclined to give it a go (implying that you are already past the barriers of: lack of familiarity with foreign markets, political risk, market efficiency, regulations, transaction costs and taxes), here I would like to talk about currency risk — what it is, how to measure it and how to hedge it.
Let’s say you are a U.S. investor looking at European stocks (listed on European exchanges). Now that the European Union is in a big mess, there must be some unreasonably beaten-up stocks that were dragged down by the market turmoil.
Since you will be buying euro-denominated assets, you will first need to get euro. This is your currency risk. Eventually, when you liquidate your European assets, you will probably want your cash converted back to dollars, unless you move to Europe and spend your money there. As an investor you are concerned with the return on your assets. As a global investor you are concerned with that and the return on the foreign currencies you are holding.
A common misconception is that adding up the return on investment in the local currency and the percentage change in the exchange rate equals the total dollar return in the domestic currency. This is very close but not quite it. The difference comes from the exchange rate effect on the capital gain which is omitted in the above calculation.
Here is the proper way to do it, followed by an illustration.
r\$ = r + s + rs
This equation calculates the US dollar return, r\$, by adding up the return on the investment in the local currency, r, the percentage change in the exchange rate, s, and the currency return that applies to the capital gain, rs.
Example:
Cost of investment (at time T)= €100
Market value of investment (at time T+1) = €120
Initial exchange rate (at time T)= \$1.20 per €
Current exchange rate (at time T+1) = \$1.50 per €
Return on investment in € (r) = (120-100)/100 = 20%
Percentage change in the exchange rate (s) = (1.50-1.20)/1.20 = 25%
Currency return on capital gain (rs) = 0.20 x 0.25 = 5%
Dollar return on investment (r\$) = 20% + 25% + 5% = 50%
Check:
Cost of investment in dollars (at time T) = €100 x \$1.20/€ = \$120
Market value of investment in dollars (at time T+1) = €120 x \$1.50/€ = \$180
Dollar return on investment = (180-120)/120 = 50%
You gained 20% on the investment and another 25% on the Euro against the dollar, plus 25% on the 20% capital gain or 5% for a total of 50% total return (dividends were not paid in this period).
As you can see, the larger the capital gain and the bigger the exchange rate movement, the more important this last part of the calculation becomes.
Another thing to keep in mind is that for the equation to hold the exchange rate should be a direct quote, meaning domestic currency per unit of foreign currency, in our case US dollars per one Euro.
Now that you know how to correctly calculate your foreign returns, let’s see what causes exchange rates to move and where they might be headed.
What Drives Exchange Rates
The short answer is: too many things for a person to know. There are just too many moving pieces in the freely floating exchange rate system. Economists have advanced many theories explaining, and trying to forecast, exchange rate movements. They are no good for precise forecasts, but they do offer some framework for thinking about exchange rates.
Two major approaches are:
The simple idea behind PPP is that if you buy a basket of goods in the USA and sell it Germany, the USD/EUR exchange rate should be such that the Euro you got should convert to the dollar amount you paid for the basket. Put differently, movements in exchange rates should offset any differences in inflation rates between two countries.
Now, this is a useful simplification but its practical value is arbitrary. As I said, there are too many moving pieces for any remotely precise forecasting to work. Nevertheless, due to mean reversion, exchange rates do tend to move towards parity over the long term (5 years and more). Of course, even if they reach a point of balance, they don’t stay there.
So, at best PPP can give you some idea about where the exchange rate is headed in the very long term. If you happen to be a long-term investor, this may be good enough for you. The OECD provides a wonderful database of comparative price levels, PPPs and exchange rates.
b) Relative economic strength (RES).
While PPP focuses on trade flows (flows of goods and services), RES focuses on investment flows. The simple idea here is that a country with healthy economic growth and attractive investment opportunities will face a strong demand for its currency. Again, this is a general theory that can give you some idea as to the direction of future exchange rate movements over the very long term.
It’s rather logical that good economic and investment climate drive capital flows and currency appreciation. For the most part of the 20th century the USA was the poster child of RES in action. The investor can combine PPP with his views on RES to get a fuller picture.
But don’t let calculating currency-adjusted returns and thinking about purchasing power parities and relative economic strengths discourage you one bit. Read on to find out why you should stop worrying and love (global) investing.
Currency Risk – Not a Barrier to International Investing
Here are several good reasons why currency risk shouldn’t prevent you from investing abroad.
a) Currency risks are not additive.
Correlations among a large number of currencies tend to cancel out. Thus, if you set out to build a truly global portfolio with assets denominated in a whole bunch of currencies, currency risk shouldn’t be of much concern to you. As currency markets will have it, when one exchange rate goes up, another will go down and overall (and over time), for your basket of currencies, these movements will cancel out.
Studies have found that currencies move erratically to such an extent that they add only some 10% to the risk of investments in a diversified global portfolio as measured by standard deviation of returns (acknowledging that this is not the favorite measure of risk of value investors).
It goes without saying that currency risk should be measured for the whole portfolio rather than for individual securities or markets, otherwise you would be ignoring the diversification benefits of holding multiple currencies.
b) Currency risk is lower for long-term investors.
The contribution of currency risk to overall portfolio risk decreases with the length of the investment horizon. This conclusion is based on PPP and RES and at its core is mean reversion. As value investors, you are very familiar with the concept.
Basically, just as stocks do, exchange rates tend to revert to the fundamentally justified (based on PPP, RES or a similar framework of your choice) mean over the long run. Thus, you can incorporate the exchange rate movement in your investment decision at the onset and forget about it (at least until a major change of course happens in the country in view).
c) However, if you are still worried, exchange rate risk may be hedged by:
§ Selling futures currency contracts.
Many brokers, even discount brokers, offer combined accounts from which you can trade all sorts of financial products, including futures.
The idea of futures is very simple. In our case, when making the investment in Germany, our US investor would enter a contract to deliver the Euro he will receive from liquidating his German investment at a fixed future date, at an exchange rate determined at the time of buying the German shares and entering the futures contract.
Futures are standardized, exchange-traded products. The Chicago Mercantile Exchange (CME) is the largest futures exchange in the world. For the retail investor, it offers the E-micro forex futures whose size is 1/10 that of the normal futures contract.
The contract size varies by currency pair but you can say that one E-micro has an underlying value of \$10,000-20,000 or the respective currency equivalent. Take for example the E-micro EUR/USD, ticker M6E, with a size of €12,500. Our US investor can choose to sell one such contract when he opens his position in Germany and becomes exposed to the Euro. Thus, he locks in the then current exchange rate and eliminates exchange rate risk. By selling one E-micro EUR/USD he is agreeing to deliver €12,500 at a future date (the expiration date of the contract) at the EUR/USD rate of 1.20. Hence, no matter what happens to the exchange rate in the meantime, our investor will be able to sell his Euro at \$1.20/€.
Currency risk is completely eliminated if the size of the futures contract(s) matches the amount of Euro to be delivered. However, that will rarely happen because the investor wouldn’t know how much his investment will appreciate or depreciate in the time between entering the futures contract and executing it. Still, hedging the principal can be a good starting point in limiting currency risk.
Unfortunately, E-micros are available only in a bunch of major currency pairs. Check whether the currency you are considered is covered before making any plans for hedging with futures.
The more serious problem with futures is their limited life. These contracts expire quarterly and are not much spread out in the future. The fact that the most active trading in the futures market is in the nearest expiration contract means that a futures position must be rolled over periodically to maintain appropriate market exposure, running up costs.
§ Selling currency ETFs or ETNs.
ETFs and ETNs trade on the stock exchange as normal shares. The difference between them is that the former are usually a basket of assets while the latter are a debt obligation (of a major bank, normally).
Their lack of an expiration date has made currency ETFs and ETNs the instruments of choice for indefinite-term portfolio hedging. They track different exchange rates. Our US investor may prefer to short the iPath EUR/USD Exchange Rate ETN, ticker ERO, to hedge his Euro exposure.
The principle is the same as holding long and short positions is stocks, just in this case we are talking about a long (your German, Euro-denominated investment) and a short (the ETF/ETN) position in a currency (the Euro). The idea is that the two positions will cancel out and the investor will be left only with the return on his investment, not affected by exchange rate movements.
Similar to the currency futures, currency options are traded on the CME and constitute standardized contracts which give you the right but not the obligation to buy or sell a currency at a future date, at a predetermined price.
Again, the investor needs to check the currency pairs on which options are available, the size of the underlying contracts (no options on E-micros), and the expirations.
Buying puts on the EUR/USD gives the investor the right to put his Euro at a future date, at a fixed exchange rate. Again, rolling contracts over will be the major concern for the long-term investor.
§ Borrowing in foreign currency to finance the investment
Instead of converting dollars to Euro, the investor can borrow Euro from his broker (depending on the broker). By involving the broker currency risk is removed from the principal of the investment. However, if the investment goes against the investor’s expectations, he will be left with debt of greater value than the cost of the loan. Hence, this is more of a modification of the risk than its full elimination.
In our example the investor borrows €100 at \$1.20/€ to purchase his German stock. From his US dollar deposit with the broker \$120 will be used as collateral on the loan. The broker will charge interest on the loan and (in some cases) will pay interest on the deposited cash collateral.
At the time when our investor decides to close the position, he has to repay the €100 loan to the broker first and is left with the remainder.
This is how the transaction looks if the investment appreciates and depreciates by 20%.
Exchange rate Gain in Euro Gain in dollars Relative to currency trade \$1.00/€ €20 \$20 +\$20 \$1.20/€ €20 \$24 \$0 \$1.50/€ €20 \$30 -\$30
Exchange rate Loss in Euro Loss in dollars Relative to currency trade \$1.00/€ (€20) (\$20) +\$20 \$1.20/€ (€20) (\$24) \$0 \$1.50/€ (€20) (\$30) -\$30
This is how the transaction looks if the investor instead converted dollars to Euro.
Exchange rate MV in Euro MV in dollars Cost in dollars Gain in dollars \$1.00/€ €120 \$120 \$120 \$0 \$1.20/€ €120 \$144 \$120 \$24 \$1.50/€ €120 \$180 \$120 \$60
Exchange rate MV in Euro MV in dollars Cost in dollars Loss in dollars \$1.00/€ €80 \$80 \$120 (\$40) \$1.20/€ €80 \$96 \$120 (\$24) \$1.50/€ €80 \$120 \$120 \$0
As you can see, by taking a loan the investor is removing the currency exposure of the principal amount. This way he improves his result relative to the currency conversion scenario in cases when the Euro weakens against the dollar (because he has less exposure to the Euro when it depreciates). But the result is worse when the Euro strengthens (because he has less exposure to the Euro when it appreciates).
Remember the formula for calculating currency-adjusted returns from the beginning of the article:
r\$ = r + s + rs
If instead of taking the currency conversion the investor goes for the loan approach, the formula changes to:
r\$ = r + rs
Basically, the percentage change in the exchange rate in this equation affects only the capital gain and not the principal amount.
I hope to have shed some light on the currency risk investors face when investing abroad — how to think about it and how to limit it so it is not an impediment to taking advantage of market opportunities around the world anymore.
dgenchev
Dimitar Genchev is a student of value.
Visit dgenchev's Website
Currently 2.63/512345 Rating: 2.6/5 (8 votes)
Batbeer2 - 5 years ago
Michelin
McDonalds
Canon
Nestle
Question:
1) Supposing I wanted to buy any of these, what currency should I use in order to minimize currency risk ?
2) Say I'm long Canon. I bought shares on the Tokio exchange. Am I safer shorting the Yen ?
Dgenchev - 5 years ago Report SPAM
Hi Batbeer2,
I see what you are driving at.
Obviously, I was talking about hedging the risk of the currency in which the shares are denominated, i.e., your portfolio's direct exposure to currency risk, not the company's exposure to international markets. The multinational concerns that you are pointing out certainly have exposures to many different currencies, but they are the concern of the CFO.
The investor could hedge or not hedge, depending on his exchange rate views or lack thereof, his exposure to the EUR (Michelin), CHF (Nestle), USD (McDonald's), JPY (Canon) respective to his domestic currency.
I should have mentioned that US investors using ADRs are also exposed to the currency fluctuations of the share price in its local market, even though ADRs are priced in USD, because ADRs are basically the shares of a foreign company held by a US custodian.
As to your second question, do you have views on the future of the yen? What are they? Where are you based? If I were a long-term US investor and based my views about the yen on PPP, I'd hedge my Canon position because I expect the yen to depreciate by over 1/3 against the dollar in the long run.
Also, this is not necessarily for you Batbeer2, I consider it common decency for the people not satisfied with an article and ranking it low to share their criticism for everyone's benefit.
Batbeer2 - 5 years ago
Hi dgenchev.
>> Also, this is not necessarily for you Batbeer2, I consider it common decency for the people not satisfied with an article and ranking it low to share their criticism for everyone's benefit.
>> As to your second question, do you have views on the future of the yen? What are they? Where are you based? If I were a long-term US investor and based my views about the yen on PPP, I'd hedge my Canon position because I expect the yen to depreciate by over 1/3 against the dollar in the long run.
I don't have a clue. IMO it's irrelevant what currency you use to pay for a given stock. I think of it like eBay. If I bid for say.... a lathe, I don't care whether the seller is in india or Canada. Once I own the machine, it no longer matters. What matters is whether it's cheap and fit for use. If so, I can recoup my investment quickly.
The question was a feeble attempt to point out that Canon shares don't know their owner. If they don't, they can't become riskier because their owner lives outside Japan.
I'm Dutch. Am I better off shorting the dollar if I'm long Lexmark ? I don't think so.
In short, buying stocks is a solution not a cause of currency risk.
Dgenchev - 5 years ago Report SPAM
Let's take a pure example, not a gray area case. Forget about the multinationals and focus on local businesses.
Buying a foreign currency denominated asset, the investor exposes himself to currency risk. If we could buy two similar assets with the same earning power, they should be worth the same no matter where they are located.
However, it does matter where they are located because different countries have different interest rates and inflation rates which are reflected in the exchange rate of their currencies. They also are reflected in the price of the asset because they spell different economic climate for the respective assets. It is a matter of economic dispute whether the exchange rate and the investment return move in the same or opposite direction.
The consensus is that in developed countries they move in opposite directions. If poor economic conditions bring about a depreciating currency, this is good for business (especially export oriented business) and drives up its competitiveness and hence, value.
This aside, the same two assets earning, say, 10% annually will leave the investor with a completely different currency-adjusted return if in one case, the investor earns another 10% because the foreign currency appreciated, while in the other case, he loses 10% because of the currency movement.
Of course, this is only realized once the investor sells the position and converts the money back to his base currency. In the meantime, he may not bother about it, because the asset is earning the 10% he is aiming for. But in the end, he will need to convert back to his base currency. Does he want to let exchange rate fluctuations take care of themselves or does he prefer to lock in a certain rate?
I think you tried to make the following point. If we consider Nestle, which is listed in Switzerland, but, for simplicity's sake let's assume, generates 99% of its revenue in the EU. Is the US investor exposed to the Swiss Frank or to the Euro? The assets will probably be in the EU. Sales and earnings will depend on the EU consumers. Capital expenditures and (partly) raw materials will be in Euro.
In this case the stock price will be much more sensitive to movements in the Euro than in the Frank. So, it won't make sense to hedge the Frank.
In the case of multinationals, you have exposures to a bunch of currencies because of the many markets the company operates in, the different suppliers and asset bases. But these exposures are for the CFO to ponder. For the investor it would be too much work and too much uncertainty to try and estimate these exposures, adjust for the hedges already in place and, based on this, decide whether he should hedge his position or not. However, it is easy to see that he has some exposure to the foreign currency in which the stock is denominated and hedge that part of his exposure.
If you think Canon's exposure to the Yen is negligible, then don't bother hedging. If you think it has some exposure but the Yen will appreciate against the Euro, let it work for you. If you think the company will generate good returns, but they might be hampered by unfavourable exchange rate movements, lock in your rate and get the pure return.
There are many ways to skin that proverbial cat :o)
P.S. Cheers to the Netherlands. I did my Bachelor's in Groningen.
Batbeer2 - 5 years ago
Hi dgenchev,
Monday, european leaders will announce they have solved the problem.
Everyone gets EUR 100 000. The Swiss decide to join the monetary union.
Merkel announces there's also a new and improved currency the "new euro". Old euros will be exchanged for new ones at a rate of 1:10.
Nestle issues a press release: "No news, we are open for business"
US investors, who were long Nestle and short the euro are glad.
So are US investors who were simply long nestle.
Dgenchev - 5 years ago Report SPAM
Are they?
The initial investment of EUR 100 is now EUR 1,000 because of the devaluation.
1 Euro buys \$0.15, not \$1.5.
The unhedged American sells his position of EUR 1,000 and gets \$150, which is exactly what he invested, for a return of 0%.
The hedged American sells the EUR 1,000 at the locked-in rate of 1.50 for a 900% profit.
Batbeer2 - 5 years ago
>> The hedged American sells the EUR 1,000 at the locked-in rate of 1.50 for a 900% profit.
I don't think so. His broker tells him he can close his short position in "new euros" 10x the amount.
Practical example: If you were short LVLT earlier this year @ 2.00, you are NOT in trouble today.
Dgenchev - 5 years ago Report SPAM
Sure.
But splits are not nearly as common for currencies as they are for stocks. Instead of having a one-off 10 for 1 split, chances are the currency will gradually depreciate and there will be no grounds for the broker not to honor the agreed upon rate.
Batbeer2 - 5 years ago
Yes
Maybe I should just short some currency and wait.
Come to think of it, I already am. I have a mortgage.
Dgenchev - 5 years ago Report SPAM
There you go!
Done and done.
That mortgage will be gone in no time :o)
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# Physical Layer
Document Sample
Physical Layer:
Signals, Capacity, and Coding
CS 4251: Computer Networking II
Nick Feamster
Spring 2008
This Lecture
• What’s on the wire?
– Frequency, Spectrum, and Bandwidth
• How much will fit?
– Shannon capacity, Nyquist
• How is it represented?
– Encoding
Digital Domain
• Digital signal: signal where intensity maintains
constant level for some period of time, and then
changes to some other level
– Amplitude: Maxumum value (measured in Volts)
– Frequency: Rate at which the signal repeats
– Phase: Relative position in time within a single period
of a signal
– Wavelength: The distance between two points of
corresponding phase ( = velocity * period)
Any Signal: Sum of Sines
• Our building block:
Asin( x
• Add enough of them to
get any signal f(x) you
want!
• How many degrees of
freedom?
• What does each
control?
• Which one encodes the
coarse vs. fine structure
of the signal?
Fourier Transform
• Continuous Fourier transform:
F(k ) F f ( x)
2ik x
f ( x) e dx
• Discrete Fourier transform:
n 1
Fk f x e
2i k x
n
x 0
• F is a function of frequency – describes how much of
each frequency f contains
• Fourier transform is invertible
Skipping a Few Steps
• Any square wave with amplitude 1 can be
represented as:
Spectrum and Bandwidth
• Any time domain signal can be represented in
terms of the sum of scaled, shifted sine waves
• The spectrum of a signal is the range of
frequencies that the signal contains
– Most signals can be effectively represented in finite
bandwidth
• Bandwidth also has a direct relationship to data
rate…
Relationship: Data Rate and Bandwidth
• Goal: Representation of square wave in a form
that receiver can distinguish 1s from 0s
• Signal can be represented as sum of sine waves
• Increasing the bandwidth means two things:
– Frequencies in the sine wave span a wider spectrum
– “Intervals” in the original signal occur more often
• [Include representation of square wave as sum
of sine waves here. Derive data rate from
bandwidth.]
Analog vs. Digital Signaling
• Analog signal: Continuously varying EM wave
• Digital signal: Sequence of voltage pulses
Signal
Analog Digital
Signal occupies same Codec produces
Analog spectrum as analog bitstream
Data data
Digital data encoded Signal consists of two
Digital using a modem voltage levels
Transmission Impairments
• Attenuation
– The strength of a signal falls off with distance over
any transmission medium
• Delay distortion
– Velocity of a signal’s propagation varies w/ frequency
– Different components of the signal may arrive at
different times
• Noise
Attentuation
• Signal strength attentuation is typically
expressed as decibel levels per unit distance
• Signal must have sufficient strength to be:
– Stronger than the noise in the channel to be received
without error
• Note: Increasing frequency typically increases
attentuation (often corrected with equalization)
Sources of Noise
• Thermal noise: due to agitation of electrons,
function of temperature, present at all
frequencies
• Intermodulation noise: Signals at two different
frequencies can sometimes produce energy at
the sum of the two
• Crosstalk: Coupling between signals
Channel Capacity
• The maximum rate at which data can be transmitted over
a given communication path
• Relationship of
– Data rate: bits per second
– Bandwidth: constrained by the transmitter, nature of
transmission medium
– Noise: depends on properties of channel
– Error rate: the rate at which errors occur
• How do we make the most efficient use possible of a
given bandwidth?
– Highest data rate, with a limit on error rate for a given bandwidth
Nyquist Bandwidth
• Consider a channel that has no noise
• Nyquist theorem: Given a bandwidth B, the
highest signal rate that can be carried is 2B
• So, C = 2B
– But (stay tuned), each signal element can represent
more than one bit (e.g., suppose more than two signal
levels are used)
– So … C = 2B lg M
• Results follow from signal processing
– Shannon/Nyquist theorem states that signal must be
sampled at twice its highest rate to avoid aliasing
Shannon Capacity
• All other things being equal, doubling the
bandwidth doubles the data rate
– Increasing the data rate means “shorter” bits
– …which means that a given amount of noise will
corrupt more bits
– Thus, the higher the data rate, the more damage that
unwanted noise will inflict
Shannon Capacity, Formally
• Define Signal-to-Noise Ratio (SNR):
– SNR = 10 log (S/N)
• Then, Shannon’s result says that, channel
capacity, C, can be expressed as:
– C = B lg (1 + S/N)
• In practice, the achievable rates are much lower,
because this formula does not consider impulse
noise or attenuation
Example
• Bandwidth: 3-4MHz
• S/N: 250
• What is the capacity?
• How many signal levels required to achieve the
capacity?
Modulation
• Baseband signal: the input
• Carrier frequency: chosen according to the
transmission medium
• Modulation is the process by which a data
source is encoded onto a carrier signal
• Digital or analog data can be modulated onto
digital and analog signals
Data Rate vs. Modulation Rate
• Data rate: rate, in bits per second, that a signal
is transmitted
• Modulation rate: the rate at which the signal
level is changed (baud)
Digital Data, Digital Signals
• Simplest possible scheme: one voltage level to
“1” and another voltage level to “0”
• Many possible other encodings are possible,
with various design considerations…
Aspects of a Signal
• Spectrum: a lack of high-frequency components
means that less bandwidth is required to
transmit the signal
– Lack of a DC component is also desirable, for various
reasons
• Clocking: Must determine the beginning and
end of each bit position.
– Not easy! Requires either a separate clock lead, or
time synchronization
• Error detection
• Interference/Noise immunity
• Cost and complexity
• Level: A positive constant voltage represents
one binary value, and a negative contant voltage
represents the other
– In the presence of noise, may be difficult to
distinguish binary values
– Synchronization may be an issue
Improvement: Differential Encoding
– Zero: No transition at the beginning of an interval
– One: Transition at the beginning of an interval
– Since bits are represented by transitions, may be
more resistant to noise
– Clocking still requires time synchronization
Biphase Encoding
• Transition in the middle of the bit period
– Transition serves two purposes
• Clocking mechanism
• Data
• Example: Manchester encoding
– One represented as low to high transition
– Zero represented as high to low transition
Aspects of Biphase Encoding
– Synchronization: Receiver can synchronize on the
predictable transition in each bit-time
– No DC component
– Easier error detection
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Numerical Ability
Numerical Ability
1. From a circular sheet of paper of radius 30 cm, a sector of 10% area is removed. If the remaining part is used to make a conical surface, then the ratio of the radius and height of the cone is __
1. 90% of area of sheet = Cross sectional area of cone
⇒ 0.9 × π × 30 × 30 = π × r1 × 30
⇒ 27 cm = r1
∴ height of the cone =√302 - 272= 13.08 cm
Correct Option: C
90% of area of sheet = Cross sectional area of cone
⇒ 0.9 × π × 30 × 30 = π × r1 × 30
⇒ 27 cm = r1
∴ height of the cone =√302 - 272= 13.08 cm
1. A coin is tossed thrice. Let X be the event that head occurs in each of the first two tosses. Let Y be the event that a tail occurs on the third toss. Let Z be the event that two tails occurs in three tosses. Based on the above information, which one of the following statements is TRUE?
1. x = {HHT, HHH}
y depends on x
z = {TTH, TTT}
∴ ‘d’ is the correct choice.
Correct Option: B
x = {HHT, HHH}
y depends on x
z = {TTH, TTT}
∴ ‘d’ is the correct choice.
1. Right triangle PQR is to be constructed in the xy– plane so that the right angle is at P and line PR is parallel to the-axis. The x and y coordinates of P, Q, and R are to be integers that satisfy the inequalities: -4 <, x < 5 and 6 < y < 16. How many different triangles could be constructed with these properties?
1. We have the rectangle with dimensions 10 ×11 (10 horizontal dots and 11 vertical). PQ is parallel to y-axis and PR is parallel to x-axis.
Choose the (x, y) coordinates for vertex P (right angle): 10C1 × 11C1
Choose the x coordinate for vertex R (as y coordinate is fixed by A): 9C1, (10 – 1 = 9 as 1 horizontal dot is already occupied by A)
Choose the y coordinate for vertex Q (as x coordinate is fixed by A): 10C1, (11 – 1 = 10 as 1 vertical dot is already occupied by A).
Hence, required number of triangles will be 10C1 × 11C1 × 9C1 × 10C1 = 9900.
Correct Option: C
We have the rectangle with dimensions 10 ×11 (10 horizontal dots and 11 vertical). PQ is parallel to y-axis and PR is parallel to x-axis.
Choose the (x, y) coordinates for vertex P (right angle): 10C1 × 11C1
Choose the x coordinate for vertex R (as y coordinate is fixed by A): 9C1, (10 – 1 = 9 as 1 horizontal dot is already occupied by A)
Choose the y coordinate for vertex Q (as x coordinate is fixed by A): 10C1, (11 – 1 = 10 as 1 vertical dot is already occupied by A).
Hence, required number of triangles will be 10C1 × 11C1 × 9C1 × 10C1 = 9900.
1. Michael lives 10 km away from where I live. Ahmed lives 5 km away and Susan lives 7 km away from where I live. Arun is farther away than Ahmed but closer than Susan from where I live. From the information provided here, what is one possible distance (in km) at which I live from Arun's place?
1. I = I live
AH = Ahmed lives
M = Michel lives
S = Susan lives
A = Arun lives
Correct Option: C
I = I live
AH = Ahmed lives
M = Michel lives
S = Susan lives
A = Arun lives
1. The binary operation – is defined as a– b = ab + (a + b), where a and b are any two real numbers. The value of the identity element of this operation, defined as the numberx such that a– x = a, for any a, is ______.
1. The binary operation – is defined
⇒ a – b = ab + (a + b)
a – x = a
∴ From the equation ‘b’ is the variable
Option A: x = 0
a – o = a × 0 + (a + 0) = 0 + a = a
Option B: x = 1
a – 1 = a × 1 + (a + 1) = a + a + 1 = 2a + 1
Option C: x = 2
a – 2 ⇒ a × 2 + (a + 2) = 2a + a + 2 = 3a+ 2
Option D: x = 10
a – 10 ⇒ a × 10 + (a + 10) = 10a + a + 10 = 11a + 10
∴ Option ‘A’ only True.
Correct Option: A
The binary operation – is defined
⇒ a – b = ab + (a + b)
a – x = a
∴ From the equation ‘b’ is the variable
Option A: x = 0
a – o = a × 0 + (a + 0) = 0 + a = a
Option B: x = 1
a – 1 = a × 1 + (a + 1) = a + a + 1 = 2a + 1
Option C: x = 2
a – 2 ⇒ a × 2 + (a + 2) = 2a + a + 2 = 3a+ 2
Option D: x = 10
a – 10 ⇒ a × 10 + (a + 10) = 10a + a + 10 = 11a + 10
∴ Option ‘A’ only True.
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## Question
### Comprehension
Statement I- There are 220 students studying in college A and college B. Students from 3 cities i.e Patna, Delhi and Mumbai study in these 2 colleges.
Statement II- Ratio of total number of students in college A and total number of students in college B is 6 : 5. Number of students of Mumbai who studies in college B is 30% of total number of students of college B.
Statement III- Number of students of Mumbai who studies in college B is 75% of number of students of Patna who studies in college A.
Statement IV- Number of students from Delhi who studies in college B is 62.5% of the number of students from Patna who studies in college A. Ratio of students from Patna in college B and number of students from Mumbai who studies in college A is 9 : 10.
# Nymber of students from Patna in college A is what % of total number of students in college B?
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1. 40%
2. 45%
3. 35%
4. 60%
5. None of these
Option 1 : 40%
## Detailed Solution
From the Statement 1
Total student in college A and B = 220
Statement 2
Ratio of student in college A and B = 6 : 5
So, Number of student in college A = (220/11) × 6 = 120
Number of student in college B = 220 - 120 = 100
Number of student in college B from Mumbai = 30% of 100 = 30
Statement 3
Number of student of Mumbai in college B = 75% of Number of student of Patna in college A
Number of student of Patna in college A = (30/75) × 100 = 40
Required Percentage = (40)/100 × 100 = 40%
∴ Number of students from Patna in college A is 40% of total number of students in college B
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# GED Math : Single-Variable Algebra
## Example Questions
### Example Question #21 : Solving For The Variable
Which of the following makes this equation true:
Possible Answers:
Correct answer:
Explanation:
To answer the question, we will solve for x, So, we get
### Example Question #21 : Solving For The Variable
Solve for b.
Possible Answers:
Correct answer:
Explanation:
To solve for b, we want b to stand alone. So, we get
### Example Question #23 : Solving For The Variable
Which of the following makes this equation true:
Possible Answers:
Correct answer:
Explanation:
To answer this, we will solve for y. So, we get
### Example Question #24 : Solving For The Variable
Which of the following makes this equation true:
Possible Answers:
Correct answer:
Explanation:
To answer this question, we will solve for y. We get
### Example Question #71 : Algebra
Which of the following makes this equation true:
Possible Answers:
Correct answer:
Explanation:
To answer the question, we will solve for x. So, we get
### Example Question #71 : Algebra
Solve for k.
Possible Answers:
Correct answer:
Explanation:
To solve for k, we want k to stand alone. So, we get
### Example Question #72 : Algebra
Solve for :
Possible Answers:
Correct answer:
Explanation:
Multiply both sides by the least common denominator, four.
This will eliminate the fractions.
Subtract on both sides.
The answer is:
### Example Question #28 : Solving For The Variable
Which of the following makes this equation true:
Possible Answers:
Correct answer:
Explanation:
To answer the question, we will solve for y. We get
### Example Question #73 : Algebra
Which of the following makes this equation true:
Possible Answers:
Correct answer:
Explanation:
To answer the question, we will solve for a. We get
### Example Question #74 : Algebra
Which of the following makes this equation true:
Possible Answers:
Correct answer:
Explanation:
To answer this question, we will solve for y. We get
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Post subject: Functional analysisPosted: Thu Jul 14, 2016 6:17 am
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Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
$\displaystyle{1)}$ We define $\displaystyle{T:\left(C\left(\left[0,1\right]\right),||\cdot||_{\infty}\right)\longrightarrow \left(\mathbb{R},|\cdot|\right)}$ by
$\displaystyle{T(f)=\int_{0}^{1}f(t)\,\mathrm{d}t}$ .
Prove that the function $\displaystyle{T}$ is a bounded linear operator and find its norm.
$\displaystyle{2)}$ Let $\displaystyle{T:\left(\mathbb{l_{2}}\,(\mathbb{N}),||\cdot||_{2}\right)\longrightarrow \left(\mathbb{l_{2}}\,(\mathbb{N}),||\cdot||_{2}\right)}$
$\displaystyle{T\,(\left(a_{n}\right)_{n\in\mathbb{N}})=\left(0,a_1,a_2,...,a_{n},a_{n+1},...\right)}$.
Prove that $\displaystyle{T}$ is a linear isometry.
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Post subject: Re: Functional analysisPosted: Thu Jul 14, 2016 6:18 am
Team Member
Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
$\displaystyle{1)}$ Let $\displaystyle{f\in C\,\left(\left[0,1\right]\right)}$. The function $\displaystyle{f}$ is Riemann integrable at $\displaystyle{\left[0,1\right]}$
as continuous at this interval, so $\displaystyle{\int_{0}^{1}f(t)\,\mathrm{d}t\in\mathbb{R}}$.
Therefore, the function $\displaystyle{T}$ is well defined.
Let $\displaystyle{f\,,g\in C\,\left(\left[0,1\right]\right)}$ and $\displaystyle{a\in\mathbb{R}}$.
Using the properties of $\displaystyle{\rm{Riemann}}$ integral, we get :
\displaystyle{\begin{aligned} T(f+g)&=\int_{0}^{1}\left(f+g\right)\,(t)\,\mathrm{d}t\\&=\int_{0}^{1}\left(f(t)+g(t)\right)\,\mathrm{d}t\\&=\int_{0}^{1}f(t)\,\mathrm{d}t+\int_{0}^{1}g(t)\,\mathrm{d}t\\&=T(f)+T(g)\end{aligned}}
and
$\displaystyle{T(a\,f)=\int_{0}^{1}(a\,f)(t)\,\mathrm{d}t=\int_{0}^{1}a\,f(t)\,\mathrm{d}t=a\,\int_{0}^{1}f(t)\,\mathrm{d}t=a\,T(f)}$
and thus the function $\displaystyle{T}$ is $\displaystyle{\mathbb{R}}$ - linear.
Let $\displaystyle{f\in C\,\left(\left[0,1\right]\right)}$. Then :
$\displaystyle{\forall\,t\in\left[0,1\right]: \left|f(t)\right|\leq \sup\,\left\{\left|f(t)\right|: 0\leq t\leq 1\right\}=||f||_{\infty}}$
so :
$\displaystyle{\int_{0}^{1}\left|f(t)\right|\,\mathrm{d}t\leq \int_{0}^{1}||f||_{\infty}\,\mathrm{d}t=||f||_{\infty}}$ and :
$\displaystyle{\left|T(f)\right|=\left|\int_{0}^{1}f(t)\,\mathrm{d}t\right|\leq \int_{0}^{1}\left|f(t)\right|\,\mathrm{d}t\leq ||f||_{\infty}}$.
In conclusion, $\displaystyle{\left|T(f)\right|\leq ||f||_{\infty}\,,\forall\,f\in C\,\left(\left[0,1\right]\right)}$, which means that $\displaystyle{T}$
is bounded, and since :
$\displaystyle{||T||=\sup\,\left\{\left|T(f)\right|: f\in C\,\left(\left[0,1\right]\right)\,,||f||_{\infty}\leq 1\right\}}$, we have that
$\displaystyle{||T||\leq 1}$. Now, the function $\displaystyle{f:\left[0,1\right]\longrightarrow \mathbb{R}\,,f(t)=1}$ is continuous, so :
$\displaystyle{f\in C\,\left(\left[0,1\right]\right)}$ and also :
$\displaystyle{||f||_{\infty}=\sup\,\left\{\left|f(t)\right|: 0\leq t\leq 1\right\}=\sup\,\left\{1\right\}=1}$
$\displaystyle{\left|T(f)\right|=\left|\int_{0}^{1}f(t)\,\mathrm{d}t\right|=\left|\int_{0}^{1}\mathrm{d}t\right|=\left|1\right|=1}$ .
Therefore, $\displaystyle{||T||=1}$ .
$\displaystyle{2)}$ Let $\displaystyle{\left(a_{n}\right)_{n\in\mathbb{N}}\in \rm{{l}_{2}}\,(\mathbb{N})}$ . The map $\displaystyle{T}$
corresponds the sequence $\displaystyle{\left(a_{n}\right)_{n\in\mathbb{N}}}$ with the sequence $\displaystyle{\left(b_{n}\right)_{n\in\mathbb{N}}}$, where
$\displaystyle{b_{n}=\begin{cases} 0\,\,\,\,\,\,\,\,\,\,\,,n=1\\ a_{n-1}\,\,\,,n\geq 2 \end{cases}}$
It's known that $\displaystyle{||a_{n}||_{2}=\sqrt{\sum_{n=1}^{\infty}a_{n}^2}\in\mathbb{R}\cap\left[0,+\infty\right)}$.
and then:
\displaystyle{\begin{aligned} ||b_{n}||_{2}&=\sqrt{\sum_{n=1}^{\infty}b_{n}^2}\\&=\sqrt{\sum_{n=2}^{\infty}a_{n-1}^2}\\&=\sqrt{\sum_{n=1}^{\infty}a_{n}^2}=||a_{n}||_{2}\in\mathbb{R}\cap\left[0,+\infty\right)\end{aligned}} .
So, the map $\displaystyle{T}$ is well defined and we observe that
$\displaystyle{||T\,\left(\left(a_{n}\right)_{n\in\mathbb{N}}\right)||_{2}=||\left(a_{n}\right)_{n\in\mathbb{N}}||_{2}\,,\forall\,\left(a_{n}\right)_{n\in\mathbb{N}}\in \rm{l_{2}}\,(\mathbb{N})\,\,(I)}$ .
If $\displaystyle{\left(a_{n}\right)_{n\in\mathbb{N}}\,,\left(b_{n}\right)_{n\in\mathbb{N}}\in \rm{l_{2}}\,(\mathbb{N})}$ and $\displaystyle{c\in\mathbb{R}}$, then :
\displaystyle{\begin{aligned} T\,\left(\left(a_{n}\right)_{n\in\mathbb{N}}+\left(b_{n}\right)_{n\in\mathbb{N}}\right)&=T\,\left(\left(a_{n}+b_{n}\right)_{n\in\mathbb{N}}\right)\\&=\left(0,a_{1}+b_{1},a_{2}+b_{2},...,a_{n}+b_{n},a_{n+1}+b_{n+1},...\right)\\&=\left(0,a_{1},a_{2},...,a_{n},a_{n+1},...\right)+\left(0,b_{1},b_{2},...,b_{n},b_{n+1},...\right)\\&=T\,\left(\left(a_{n}\right)_{n\in\mathbb{N}}\right)+T\,\left(\left(b_{n}\right)_{n\in\mathbb{N}}\right)\end{aligned}}
\displaystyle{\begin{aligned} T\,\left(c\,\left(a_{n}\right)_{n\in\mathbb{N}}\right)&=T\,\left(\left(c\,a_{n}\right)_{n\in\mathbb{N}}\right)\\&=\left(0,c\,a_1,c\,a_2,...,c\,a_n,c\,a_{n+1},...\right)\\&=c\,\left(0,a_1,a_2,...,a_n,a_{n+1},...\right)\\&=c\,T\,\left(\left(a_{n}\right)_{n\in\mathbb{N}}\right)\end{aligned}}
so, the function $\displaystyle{T}$ is $\displaystyle{\mathbb{R}}$ - linear and according to $\displaystyle{(I)}$ is bounded and isometry.
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# suppose a map is scaled so that a half inch represents 25 miles. how many inches of the map represents 87 1/2 miles?
35,906 questions, page 31
1. ## Math
What is the surface area of a cube whose sides measure 5 inches? I did A= s squared A=5 squared 25 inches squared Then SA= 25*6=150 inches squared Is this correct? Thank you
asked by B6 on March 16, 2013
2. ## math
The scale drawing of a swimming pool measures 2 inches by 3 inches. On the drawing .25 inches = 3 feet. What are the actual dimensions of the pool?
asked by yeeh on March 7, 2016
3. ## math
Carol wants to make a quilt using a scale of 1.5 inches = 2 feet. Her scale drawing is 7.5 inches by 10.5 inches. What is the area of the actual quilt?
asked by jakira on May 16, 2016
4. ## math
If a right triangle has a base of 4 inches and a height of 3 inches, make an equivilant triangle with a base of 17 inches. Can someone explain how to figure this answer?
asked by Cm on January 25, 2010
5. ## algebra
Courtney's dog sleeps in a crate that is 40 inches wide, 30 inches tall, and 44 inches deep. What is the volume of the dog crate?
asked by Shea on March 4, 2012
6. ## algebra
Find the length of one base of a trapezoid if the area is 126 inches squared, the height is 7 inches and the length of the other base is 15 inches.
asked by Anonymous on October 8, 2011
7. ## algebra
1. Courtney's dog sleeps in a crate that is 24 inches wide, 30 inches tall, and 40 inches deep. What is the volume of the dog crate?
asked by Reagan on February 6, 2012
8. ## Lost in the pits of ALG
I have 5 problems I seem to not be able to comprehend out of 150 that are due Tuesday. I have tried every which way to solve them on my own..please help. Find the variation constant in which y varies inversely as x, and the following condition exists. Y=30
asked by TOTALLY lost on March 11, 2012
9. ## Geology
One of the curious things about the loss of life in regions surrounding the Bay of Bengal is that some 38,195 lives were lost in Sri Lanka, whereas only 2 lives were lost in Bangladesh. How could this be? It certainly couldn’t be the difference in
asked by Anonymous on February 27, 2010
10. ## Math
A carton can hold 1,000 unit cubes that measures 1inch by 1 inch by 1 inch. Describe the dimensions of the carton using unit cubes? I don't understand the question
asked by Joshua G on March 27, 2017
11. ## Math
A tabletop is made of two layers of plywood. The top layer is .584 inch thick. The bottom layer is .258 inch thick. What is the total thickness of the tabletop?
asked by DestinyLashea on February 24, 2011
12. ## math
a figure is made up of two identical triangles a 15 inch by 12 inch rectangle and a square of side 19 inchs what is the lenght of one side of each triangle if all the sides of the triangles are equal in lenght
asked by Anonymous on November 7, 2012
13. ## Math
Jose has a 120-inch roll of wrapping paper. He needs an 18-inch piece to wrap each present. Does Jose have enough to wrap 7 presents? I divided 120÷18 and got 6.66 but if I round 6.66 to 7, will the answer be yes?
asked by Bub on April 13, 2016
14. ## statistics
Distance travelled in a week by sales staff employed by nationwide security firm is approximately normally distributed with a mean of miles and standard deviation of 48 miles. Estimate the percentage of staff who in a week travel less than 80 miles
asked by Anonymous on May 3, 2010
15. ## fundamentals of math
By holding a ruler 3 feet in front of her eyes as illustrated in the diagram, Carla sees that the top and bottom points of a vertical cliff face line up with marks separated by 3.5" on the ruler. According to the map, Carla is about a half mile from the
asked by Anonymous on August 6, 2016
16. ## physics
a very strong girl runs 200 miles for several hours. After a brief rest she runs another 100 miles north. She stops for the night at a local bed and breakfast and the next morning ealks 75 miles directly south-east. Find the magnitude and direction of her
asked by idk 22 on September 28, 2010
17. ## Math
Hi I need a little help with a couple math questions I believe I have the answers, if someone can please just check them thank you. 1. The table shows the number of miles driven over time. Time 4|6|8|10 Miles 204|306|408|510 Express the relationship
asked by Anonymous on December 6, 2016
18. ## Algebra 2
A commercial freight carrier is flying at a constant speed of 400 mi/h and is traveling 4 miles east for every 3 miles north. A private plane is observed to be 210 miles east of the commercial carrier, traveling 12 miles north for every 5 miles west. a. If
asked by newt on September 28, 2009
19. ## MAth algebra 1
Steve is driving to a ski resort. At first, he drove at a rate of 65 mph. Then a snow storm started and he had to slow down to 40 mph. In total. it took him 6 hours to drive 310 miles. Which system of equations represents this situation? A. 6x + y= 1 x +
asked by dbzaltheway on February 8, 2016
20. ## Mathematics
A space camera circles the Earth at a height of h miles above the surface. Suppose that d distance, IN MILES, on the surface of the Earth can be seen from the camera. (a) Find an equation that relates the central angle theta to the height h. (b) Find an
asked by Javier on February 22, 2008
21. ## math
Micah walks on a treadmill at 4 miles per hour. He has walked 2 miles when Luke starts running at 6 miles per hour on the treadmill next to him. If their rates continue, will Luke's distance ever equal Micah's distance? Explain. *At 5 hours Luke will run
asked by :) on February 3, 2017
22. ## Physics
A bus started out going 20 miles per hour on the highway. One hour later it drove 40 miles per hour on the freeway. How long did it take to go 140 miles?
asked by Anonymous on March 8, 2012
23. ## Math, Pleas help.
Please help me with the questions provided below; I have studied in Math, but I want to make sure my answers are accurate to ace. 1.) Solve the proportion. 5/8= y/40. A.) 25 B.) 13 C.) 11 D.) 10 I chose answer choice A for number one since 8x5=40, and
asked by Anonymous on February 17, 2016
24. ## Statistics and Probability
A computer supply house receives a large shipment of blank CDs each week. Past experience has shown that the number of flaws per CD can be described by the following probability distribution, where the random variable X represents the number of flaws per
25. ## chemistry
suppose a new car travels 70.0 miles on one gallon of gasoline with 40% efficient engine. how much fuel is wasted annually due to energy efficiency if in one yr, 478.26 gallons of fuel is burned?
asked by LISA on February 10, 2008
26. ## algebra
Seven inches are equivalent to 17.78 cm. Find the equation converting length in cm to length in inches. What is the length in inches of a 96.2 cm bar? 17.8cm =7 in. 1cm = (7/17.78)in 96.2cm = 96.2 (7/17.78)in 96.2cm = 37.874 in can this be graphed and how?
asked by p eg on February 26, 2011
27. ## math
Sophina bought 3 yards of trim to put around a rectangular scarf. She does not want the scarf to be wider than 12 inches or narrower than 6 inches. Using whole numbers, if she uses all the trim, what are the possible dimensions of her scarf in inches?
asked by Blake on March 30, 2011
28. ## Math
The volume of a suitcase is 1,200 cubic inches. Its length is 20 inches, and its height is 10 inches. What is the width? Write an equation to represent the width. Then find the width of the suitcase.
asked by Anonymous on May 1, 2018
29. ## Math. 2
Siphons bought 3 yards of trim to put around a rectangular scarf. She wants the width of the scarf to be a whole number that is at least 6 inches and at most 12 inches. If she uses all the trim , what are the possible dimensions of the scarf? Write in
asked by Jenny 2 on March 10, 2014
30. ## physics
On her trip from home to school, Karla drives along three streets after exiting the driveway. She drives 1.85 miles south, 2.43 miles east and 0.35 miles north. Determine the magnitude of Karla's resultant displacement. explanation on answer please
asked by hery on October 13, 2016
31. ## algebra
36 Upstream, downstream. Junior’s boat will go 15 miles per hour in still water. If he can go 12 miles downstream in the same amount of time as it takes to go 9 miles upstream, then what is the speed of the current?
asked by Linda on December 4, 2009
32. ## math
How many fewer minutes would it take the driver to travel a distance of 10 miles at a speed of 65 miles per hour than at a speed of 50 miles per hour? Round your answer to the nearest minute.
asked by mathew on November 19, 2015
33. ## Pre Calculus
A motorboat travels 48 miles down a river in the same time it takes to travel 32 miles up the river. If the rate of the current is 4 miles per hour that is the rate of the boat still water?
asked by Jody on September 15, 2014
34. ## Math
A LIGHTHOUSE IS 10 MILES NORTHEAST OF A DOCK. A SHIP LEAVES THE DOCK AT NOON,AND SAILS EAST AT A SPEED OF 12 MILES AN HOUR. AT WHAT TIME WILL IT BE 8 MILES FROM THE LIGHTHOUSE?
asked by Shihara on March 15, 2012
35. ## Algebra
Two trains start from the same place at the same time. They travel i~ opposite directions. One averages 60 miles per hour and the other averages 80 miles per hour. After how long will they be 420 miles apart?
asked by LIbby on April 2, 2012
36. ## Math related rates
Water is poured into a conical paper cup at the rate of 3/2 in3/sec (similar to Example 4 in Section 3.7). If the cup is 6 inches tall and the top has a radius of 4 inches, how fast is the water level rising when the water is 2 inches deep? I got
asked by chan on March 2, 2017
37. ## geometry 10
Type a counter-example that would have to exist in order for the conclusion to be false. Every map that has ever been drawn can be colored with four colors so that no two regions colored alike touch at more than one point. Conclusion: Every map can be
asked by amy s on March 3, 2010
38. ## Calculus
According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. A. What is the probability that an adult male chosen at random is between 61 inches
asked by Robin on November 3, 2013
39. ## Statistics
Suppose the heights of 18-year-old men are approximately normally distributed, with mean 66 inches and standard deviation 4 inches. (a) What is the probability that an 18-year-old man selected at random is between 65 and 67 inches tall? (Round your answer
asked by Amy on November 17, 2016
40. ## math
Keith is wrapping ribbon around a cylindrical vase he will need additional ribbon to make a bow the vase has a volume of 602.88 cubic inches the bow requires 3 feet of ribbon to the nearest inch what is the least amount of ribbon including the bow does
asked by tamara ivory on April 11, 2011
41. ## math
Jack swam 3.4 miles in the same time that Christine swam 4.1 miles. How many more miles did Christine swim? in miles
asked by anonymous on April 2, 2019
42. ## Math
If X=11 inches, Y=8 inches, and Z=6 inches, what is the area of the parallelogram above?
asked by Garrett on February 27, 2011
43. ## Math
In the diagram below, AB is parallel to DE . AB =20 inches ,DE=12 inches, and BC=15 inches. what is the of DC?
asked by Fahmida on May 3, 2013
44. ## Algebra
A lawyer drives from her home, located 3 miles east and 9 miles north of the town courthouse, to her office, located 4 miles west and 15 miles south of the courthouse. Find the distance between the lawyer's home and her office.
asked by Frankie on September 18, 2018
45. ## math
There are two roads that connect Emily's house and John's house. Emily told john that she drove 3/4 of the total distance from her house to john's house on one day, but she did not tell him which road she took. A. using the long road, the distance between
asked by I NEED HELP FAST!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! on May 28, 2014
46. ## Math
The height of a TV antenna is 11ft 7in. If the height of a building is 64ft 9in, find the total height of a building including the antenna. I kept trying to do this, but I never learned this so I have no idea how to do this. adding them, I get 75 ft 16
asked by Jack on July 2, 2007
47. ## Math 102
A bag contains six marbles, of which four are red and two are blue. Suppose two marbles are chosen at random and X represents the number of red marbles in the sample. Looking for the expected value of X.
asked by Sean on March 21, 2011
48. ## Physics
A trebuchet was a hurling machine built to attack the walls of a castle under siege. A large stone could be hurled against a wall to break apart the wall. The machine was not placed near the wall because then arrows could reach it from the castle wall.
asked by barich on September 25, 2015
49. ## Physics
An airplane of a certain density and shape flies at a constant speed. To do so, it must fly with a certain velocity v0. If the size of the airplane is scaled up in length, width, and height by a factor of two, it can only fly above a new velocity v1. What
asked by Carlos on June 1, 2013
50. ## algebra
A bouncy ball bounces 16 feet high on its first bounce and exactly half as high on the next bounce. For each later bounce, the ball bounces half as high as on the previous bounce. How many inches high does the ball bounce on its seventh bounce?
asked by samay on February 6, 2018
51. ## Math
Hunter and his father drove 582 miles in 4 days. They drove 102 miles on the first day. Then they drove the same number of miles on the second, third, and fourth days. How many miles did they drive each day after the first day?
asked by Vy on May 30, 2013
52. ## calculus
An open box is to be made out of a 8-inch by 14-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. Find the dimensions of the resulting box that has the largest volume. Dimensions of the bottom of
asked by casidhe on April 20, 2012
53. ## Discrete Mathematics
Which of the following scenarios could be described by the real valued function f(x) = x3 + 3x2 + 2x? A shoebox has a length that is 1 inch longer than its width and a height that is 1 inch longer than its length, We want to know the product of three
asked by Joy on July 30, 2013
54. ## math
16. Assume that there are 20,000 runners in the New York City Marathon. Each runner runs a distance of 26 miles. If you add together the total number of miles for all the runners, how many times around the globe would the marathon runners have gone?
asked by Carly on September 4, 2018
55. ## math
the water in leroy's and jerod's fish tank has evaporated so it was about 5/8 inch below the leverit should be.they added water and the water level went up about 3/4 inch.did the water level end up above or below where it should be? answer
asked by barb on March 9, 2012
56. ## Math
what is 1+one half of square,+ one half of eight,+ one half of diamond + e. plus one half of zero
asked by Sonnya on March 14, 2019
57. ## Physics - same ? as marina but im smarter, lol
Marina is my bf and we both attend a educational program called proyecto science. In it, when we were to attend the physics course, we had this ?: T = 2ð² L/G in this equation, T represents time, L represents Length and G represents Gravity. We are
asked by shivangi on July 2, 2008
58. ## Geometry
It takes 69 inches of ribbon to make a bow and wrap the ribbon around a box, as shown. The bow takes 30 inches of ribbon. The width of the box is 11 inches. What is the height of the box? Answer in units of inches.
asked by Chitra on November 27, 2009
59. ## math
if the weight of a fish varies jointly as the length and the square of the girth, and if a fish 11 inches long with a girth of 7 inches weighs 0.7lbs, find the weight of a fish 23 inches long with a girth of 18 inches.
asked by amber on September 5, 2012
60. ## math
a rope was cut in 7 equal pieces of 1/2 a piece measures 3 inch. how many inch was the rope before it was cut
asked by aaron on February 5, 2013
61. ## Math
What would be an expression that can be used to show how many lengths of 5/8 inch lengths of string can be cut from a 15 inch length of string
asked by Joe on November 18, 2010
62. ## Math
If a 2-inch pipe will fill a tank in 6 hours, how long will it require for a 3-inch pipe to fill it?
asked by Laurie on May 6, 2016
63. ## Math
Mr. Munoz has a 1/16 inch drill bit and a 3/32 inch drill bit. Which will make a bigger hole?
asked by Angel on June 27, 2012
64. ## Math
You have \\$20\$20dollar sign, 20 to spend on taxi fare. The ride costs \\$5\$5dollar sign, 5 plus \\$2.50\$2.50dollar sign, 2, point, 50 per mile. Let mmm represent the number of miles ridden. Write an inequality to determine how many miles you can ride for
asked by Anuraj on February 9, 2017
65. ## MATH HELP ASAP PLEASE!!!!!
Please, please, please, help me! Directions: A large pizza at Tony's Pizzeria is a circle with a 14-inch diameter. Its box is a rectangular prism that is 1 14 8 inches long, 1 14 8 inches wide, and 3 1 4 inches tall. Your job is to design a crazy new shape
asked by Lilitwan Doesn't Understand on April 18, 2017
66. ## algebra
mang pedring wanted to construct a square table such that the length of its side is 30 cm longer than its height. What expression represents the area of the table? Suppose the table is 90 cm high what would be the area?
asked by Sella on July 11, 2012
67. ## social studies
Do you believe that the present flag adequately represents the Virgin Islands? Why? Yes, because a flag represents an entire country. Some would argue that there are 50 stars on the flag, and none represent the Virgin Islands or even the District of
asked by Anastasia on August 27, 2006
68. ## Algebra
The table shows the number of miles driven over time. Find the rate of change. x- y- 4 | 204 6 | 306 8 | 408 10 | 510 A.) 51/1; Your car travels 51 miles every 1 hour. * B.) 204; Car travels 204 miles C.) 1/51; Car travels 51 miles every 1 hour. * The
asked by Dylan on December 13, 2017
69. ## Math
A ship leaves England for the Strait of Magellan. It is traveling at a constant speed of eight knots and takes 41 days to reach the Strait of Magellan. The ship reaches the Strait of Magellan at the exact time of day that it left England. How far did the
asked by ana on December 9, 2014
70. ## Physics
Suppose when we look in one half of the sky, the CMBR appears to be at a temperature of 2.72K. What is the peak wavelength in that direction? Are we moving toward or away from the region of space? What is our velocity with respect to the CMBR?
asked by Leila on April 23, 2012
71. ## Astronomy
Suppose when we look in one half of the sky, the CMBR appears to be at a temperature of 2.72K. What is the peak wavelength in that direction? Are we moving toward or away from the region of space? What is our velocity with respect to the CMBR?
asked by Leila on April 23, 2012
72. ## Physics
Suppose when we look in one half of the sky, the CMBR appears to be at a temperature of 2.72K. What is the peak wavelength in that direction? Are we moving toward or away from the region of space? What is our velocity with respect to the CMBR?
asked by Leila on April 23, 2012
73. ## Physics
Suppose when we look in one half of the sky, the CMBR appears to be at a temperature of 2.72K. What is the peak wavelength in that direction? Are we moving toward or away from the region of space? What is our velocity with respect to the CMBR?
asked by Leila on April 24, 2012
74. ## Astronomy
Suppose when we look in one half of the sky, the CMBR appears to be at a temperature of 2.72K. What is the peak wavelength in that direction? Are we moving toward or away from the region of space? What is our velocity with respect to the CMBR?
asked by Leila on April 23, 2012
75. ## Astronomy
Suppose when we look in one half of the sky, the CMBR appears to be at a temperature of 2.72K. What is the peak wavelength in that direction? Are we moving toward or away from the region of space? What is our velocity with respect to the CMBR?
asked by Leila on April 23, 2012
76. ## Physics
Suppose when we look in one half of the sky, the CMBR appears to be at a temperature of 2.72K. What is the peak wavelength in that direction? Are we moving toward or away from the region of space? What is our velocity with respect to the CMBR?
asked by Leila on April 24, 2012
77. ## Physics
Suppose when we look in one half of the sky, the CMBR appears to be at a temperature of 2.72K. What is the peak wavelength in that direction? Are we moving toward or away from the region of space? What is our velocity with respect to the CMBR?
asked by Leila on April 23, 2012
78. ## Chem
Suppose that you start with 1000000 atoms of a particular radioactive isotope. how many half-lives woruld be required to reduce the number of undecayed atoms to fewer than 1000?
asked by Shay on October 20, 2014
79. ## Physics
Suppose when we look in one half of the sky, the CMBR appears to be at a temperature of 2.72K. What is the peak wavelength in that direction? Are we moving toward or away from the region of space? What is our velocity with respect to the CMBR?
asked by Leila on April 24, 2012
80. ## Physics
Suppose when we look in one half of the sky, the CMBR appears to be at a temperature of 2.72K. What is the peak wavelength in that direction? Are we moving toward or away from the region of space? What is our velocity with respect to the CMBR?
asked by Leila on April 23, 2012
81. ## algebra
Please help me anybody Abby Garland is parked at a mile marker on an east -west country road.She decides to toss a fair coin 10 times ,each time driving 1 mile east if it lands heads up and 1 mile west if it lands up tails up. The term random walk applies
asked by Fatima on August 22, 2012
82. ## math
Can you add 2 hours and 45 minutes to 3 hours and 57 minutes by using decimal numbers (2.45 + 3.57)? Explain. Also, suppose you want to add 3 feet and 7 inches to 5 feet 6 inches. Can you use decimal numbers so that you can add 3.7 to 5.6 to get the total
asked by anonymous on November 18, 2013
83. ## MATH
15. Write a function rule that represents y is 5 less than the product of 4 and x. 16. Write a function rule that represents 7 less than three fifths of b is a. 17. Write a function rule that represents the almond extract a remaining in an 8 oz bottle
84. ## math
Extra Credit opportunity: A driver is headed north on a long, straight highway and sees this sign: Nearville 150 miles Farville 160 miles Then, surprisingly, an hour later she sees this apparently inconsistent sign on the same highway: Nearville 100 miles
asked by Anonymous on March 2, 2012
85. ## chemistry
An alien rocketship is traveling at a speed of 4,900,000 4{,}900{,}000 inches per hour. At this speed, how many miles will it travel in 1 minute? Round to the nearest thousandth.
asked by britney on September 5, 2012
86. ## math
Brandon and Jacob buy used cars at the same time. Brandon buys a car with 10,000 miles on it. He drives an average of 100 miles a week. The equation that follows can be used to determine how many miles, m, will be on the car after any number of weeks of
asked by lance on November 13, 2013
87. ## math
Ashwin and Donald decided to set out from two towns on their bikes, which are 247 miles apart, connected by a straight Roman road in England. When they finally met up somewhere between the two towns, Ashwin had been cycling for 9 miles a day. The number of
asked by blodwick on April 19, 2016
88. ## Math
A rectangular flowerpot is 20 inches long 8 inches wide and 6 inches high. true or false if andy fills the flowerpot 2/3 fullof potting soil the expression 2/3(20 + 8 + 6) could be used to find the amount of potting soil he needed
asked by Lisa on December 5, 2010
89. ## Algebra
A rectangular piece of cardboard is 15 inches longer than it is wide. If 5 inches are cut from each corner, and the remaining fold up to form a box,the volume of the box is 1250 cubic inches. Find the dimensions of the piece of cardboard.
asked by Alberto on May 12, 2014
90. ## Algebra
Define variables, write an equation and solve. The perimeter of a scalene triangle is 75 inches. The middle side is 6 inches longer than the shortest side. The longest side is 15 inches less than twice the middle side. Find the length of all three sides.
asked by Jasmann on November 11, 2017
91. ## math
A garbage can is in the shape of a rectangular prism. The area of the base of the garbage can is 480 square inches. The height of the garbage can is 30 inches. (Part A) how many cubic inches of space are inside this garbage can? (Part B) Explain how you
asked by tony on September 25, 2011
92. ## Math
Arthur is trying to figure out the heights of 3 of his friends. Here are the facts he know: 1. The sum of the heighta oglf these 3 friends is 17 fast 8 inches. 2. The shortest friends is 5 feet 6 inches tall 3. The other 2 friends differ in height by 4
93. ## Math
A glass box shaped like a rectangular prism has width of 7 in., length 9 in., and height 12 in. . It is being shipped in a bigger box with width 10 inches, length 10 inches, and height 15 inches. The space between the box and the shipping box will be
asked by A girl with problems. on March 18, 2016
94. ## Geometry
A cylinder flower pot with open top needs to be painted. The height is 9 inches and radius is 3 inches. If it takes Troy 5 minutes to paint 40 square inches, how long to the minute will it take him to paint the outside of the flower pot.
asked by Mary on May 23, 2011
95. ## Statistics
Team X has an average (mean) height of 70 inches and a standard deviation of 3 inches and Team Y has an average height of 67 inches with a standard deviation of 9 inches. Compare the two teams using this information and which team will have the tallest and
asked by Becky on November 8, 2015
96. ## Math
Suppose that the function represents the percentage of inbound e-mail in the U.S. that is considered spam, where x is the number of years after 2002. Use this model to approximate the percentage of spam in the year 2006 to the nearest tenth of a percent.
asked by Anna on December 10, 2008
97. ## math
A mailbox is 15 inches long and 8 inches wide. The height of the mailbox before the curve is 6 inches. What is the volume of the mailbox? Use 3.14 for pi.
asked by Rohmell on June 3, 2017
98. ## math
if robbie runs 10 miles per hour, how many minutes would it take to run 2 miles? I don't know how to figure this out? Can someone give me a hint? is it 12 minutes? 60 min/ 10 miles = x min / 2
asked by cici on December 10, 2013
99. ## Math
Two cars are 400 miles apart and start towards one another, one car traveling 4 miles per hour slower than the other. After 3 hours, the cars are 40 miles apart. Find the speed of each car
asked by Molly on October 9, 2016
100. ## math
If a man drove 3.5 miles to a gas station and on the way home drove 2 times that +.7 miles,how many miles did he drive?? 3.5 + 2 times 3.5 + .7 = what Do the math. 17.5 omg ur so smart...
asked by Caitlyn on June 7, 2007
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1. ## Re: 5th Grade Math Puzzles.
Originally Posted by Champ Kind
Puzzle 5
You say my number when you start at 0 and count by 15?s.
You say my number when you start at 0 and count by 25?s.
My number is less than 200.
My number is even.
I don't think 75 is an even number!
Here's a question my high school math teacher asked us:
You have 23 cubes of sugar. How can you put them all in two cups of coffee so that each cup of coffee has an odd number of sugar cubes in it (you can't break the cubes, although that wouldn't help).
After much deliberation and frustration by the class, she gave us the answer. She said, "Put 22 cubes in one cup and one cube in the other."
When we all protested that 22 is an even number, she calmly replied, "But isn't 22 an odd number of sugar cubes to put in a cup of coffee?"
2. ## Re: 5th Grade Math Puzzles.
I would have punched that teacher directly in the ovaries if she gave me a BS math puzzle/answer like that.
3. ## Re: 5th Grade Math Puzzles.
Originally Posted by homerdindon
I would have punched that teacher directly in the ovaries if she gave me a BS math puzzle/answer like that.
That was hilarious!
4. ## Re: 5th Grade Math Puzzles.
Originally Posted by Champ Kind
Puzzle 5
You say my number when you start at 0 and count by 15?s.
You say my number when you start at 0 and count by 25?s.
My number is less than 200.
My number is even.
I don't think 75 is an even number!
Typical test issue in my life, transcribed the wrong answer from my scratch sheet.
After hemming and hawing over the criteria - I opted for the easiest solution. 2 In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. In this example - 1 & 2.
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Engineering Courses
Engineering Mathematics Certification Exam Tests
Engineering Mathematics Practice Test 19
# Mathematical Model Quiz PDF: Questions and Answers - 19
Books:
Apps:
The Mathematical Model Quiz Questions and Answers PDF, Mathematical Model Quiz Answers PDF e-Book download Ch. 3-19 to solve Engineering Mathematics Practice Tests. Learn Introduction to Differential Equations MCQ Questions PDF, Mathematical Model Multiple Choice Questions (MCQ Quiz) to study online college courses. The Mathematical Model Trivia App: Free download learning app for concepts of solution, general rules, laplace transform introduction, laplace transform of trigonometric functions, mathematical model test prep to enroll in online classes.
The Quiz: To solve engineering problem, we have to formulate pattern as math expression in term of variables, functions and equations, such expression is called; "Mathematical Model Quiz" App (iOS & Android) with answers: Math model; Function model; Variable model; Math equation; to enroll in online classes. Study Introduction to Differential Equations Questions and Answers, Apple Book to download free sample for online high school college acceptance.
## Mathematical Model Questions and Answers : Quiz 19
MCQ 91:
To solve engineering problem, we have to formulate the pattern as math expression in term of variables, functions and equations, such expression is called
1. function model
2. math model
3. variable model
4. math equation
MCQ 92:
Laplace transform of function f(t)=cos(wt) is
1. s/(s+w)
2. s/(s-w)
3. s/(s2+w2)
4. w/(s2+w2)
MCQ 93:
Laplace transform when applied to function, changes that function into new function by using a process that involves
1. integration
2. differentiation
3. binary manipulation
4. logical manipulation
MCQ 94:
Derivative of ∂x ln(2x+3) is equals to
1. 2/(2x+3)
2. 2x/(2+3x)
3. 2x/(2x+3)
4. 1/(2x+3)
MCQ 95:
The curve of the function i.e. y=h(x) is called
1. solution curve
2. scalar curve
3. separation curve
4. summation curve
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# Logical Design.
## Presentation on theme: "Logical Design."— Presentation transcript:
Logical Design
Design with Basic Logic Gates
Logic gates: perform logical operations on input signals Positive (negative) logic polarity: constant 1 (0) denotes a high voltage and constant 0 a low (high) voltage Synchronous circuits: driven by a clock that produces a train of equally spaced pulses Asynchronous circuits: are almost free-running and do not depend on a clock; controlled by initiation and completion signals Fanout: number of gate inputs driven by the output of a single gate Fanin: bound on the number of inputs a gate can have Propagation delay: time to propagate a signal through a gate
Analysis of Combinational Circuits
Circuit analysis: determine the Boolean function that describes the circuit Done by tracing the output of each gate, starting from circuit inputs and continuing towards each circuit output Example: a multi-level realization of a full binary adder C0 = AB + (A + B)C = AB + AC + BC S = (A + B + C)[AB + (A + B)C]’ + ABC = (A + B + C)(A’ + B’)(A’ + C’)(B’ + C’) + ABC = AB’C’ + A’BC’ + A’B’C + ABC = A B C
Simple Design Problems
Parallel parity-bit generator: produces output value 1 if and only if an odd number of its inputs have value 1 P = x’y’z + x’yz’ + xy’z’ + xyz
Simple Design Problems (Contd.)
Serial-to-parallel converter: distributes a sequence of binary digits on a serial input to a set of different outputs, as specified by external control signals
Logic Design with Integrated Circuits
Small scale integration (SSI): integrated circuit packages containing a few gates; e.g., AND, OR, NOT, NAND, NOR, XOR Medium scale integration (MSI): packages containing up to about 100 gates; e.g., code converters, adders Large scale integration (LSI): packages containing thousands of gates; arithmetic unit Very large scale integration (VLSI): packages with millions of gates
Comparators n-bit comparator: compares the magnitude of two numbers X and Y, and has three outputs f1, f2, and f3 f1 = 1 iff X > Y f2 = 1 iff X = Y f3 = 1 iff X < Y f1 = x1x2y2’ + x2y1’y2’ + x1y1’ = (x1 + y1’)x2y2’ + x1y1’ f2 = x1’x2’y1’y2’ + x1’x2y1’y2 + x1x2’y1y2’ + x1x2y1y2 = x1’y1’(x2’y2’ + x2y2) + x1y1(x2’y2’ + x2y2) = (x1’y1’ + x1y1)(x2’y2’ + x2y2) f3 = x2’y1y2 + x1’x2’y2 + x1’y1 = x2’y2(y1 + x1’) + x1’y1
4-bit/12-bit Comparators
Four-bit comparator: 11 inputs (four for X, four for Y, and three connected to outputs f1, f2 and f3 of the preceding stage) 12-bit comparator:
Data Selectors Multiplexer: electronic switch that connects one of n inputs to the output Data selector: application of multiplexer n data input lines, D0, D1, …, Dn-1 m select digit inputs s0, s1, …, sm-1 1 output
Implementing Switching Functions with Data Selectors
Data selectors: can implement arbitrary switching functions Example: implementing two-variable functions
Implementing Switching Functions with Data Selectors (Contd.)
To implement an n-variable function: a data selector with n-1 select inputs and 2n-1 data inputs Implementing three-variable functions: z = s2’s1’D0 + s2’s1D1 + s2s1’D2 + s2s1D3 Example: s1 = A, s2 = B, D0 = C, D1 = 1, D2 = 0, D3 = C’ z = A’B’C + AB’ + ABC’ = AC’ + B’C General case: Assign n-1 variables to the select inputs and last variable and constants 0 and 1 to the data inputs such that desired function results
Priority Encoders Priority encoder: n input lines and log2n output lines Input lines represent units that may request service When inputs pi and pj, such that i > j, request service simultaneously, line pi has priority over line pj Encoder produces a binary output code indicating which of the input lines requesting service has the highest priority Example: Eight-input, three-output priority encoder z4 = p4p5’p6’p7’ + p5p6’p7’ + p6p7’ + p7 = p4 + p5 + p6 + p7 z2 = p2p3’p4’p5’p6’p7’ + p3p4’p5’p6’p7’ + p6p7’ + p7 = p2p4’p5’ + p3p4’p5’ + p6 + p7 z1 = p1p2’p3’p4’p5’p6’p7’ + p3p4’p5’p6’p7’ + p5p6’p7’ + p7 = p1p2’p4’p6’ + p3p4’p6’ + p5p6’ + p7
Priority Encoders (Contd.)
Decoders Decoders with n inputs and 2n outputs: for any input combination, only one output is 1 Useful for: Routing input data to a specified output line, e.g., in addressing memory Basic building blocks for implementing arbitrary switching functions Code conversion Data distribution Example: 2-to-4- decoder
Decoders (Contd.) Example: 4-to-16 decoder made of two 2-to-4 decoders and a gate- switching matrix
Decimal Decoder BCD-to-decimal: 4-to-16 decoder made of two 2-to-4 decoders and a gate- switching matrix
Decimal Decoder (Contd.)
Implementation using a partial-gate matrix:
Implementing Arbitrary Switching Functions
Example: Realize a distinct minterm at each output
Demultiplexers Demultiplexers: decoder with1 data input and n address inputs Directs input to any one of the 2n outputs Example: A 4-output demultiplexer
Seven-segment Display
Seven-segment display: BCD to seven-segment decoder and seven LEDs Seven-segment pattern and code: A = x1 + x2’x4’ + x2x4 + x3x4 B = x2’ + x3’x4’ + x3x4 C = x2 + x3’ + x4 D = x2’x4’ + x2’x3 + x3x4’ + x2x3’x4 E = x2’x4’ + x3x4’ F = x1 + x2x3’ + x2x4’ + x3’x4’ G = x1 + x2’x3 + x2x3’ + x3x4’
Sine Generators Combinational sine generators: for fast and repeated evaluation of sine Input: angle in radians converted to binary Output: sine in binary z1 = x1’x2 + x1x2’ + x2x3’ + x1’x3x4 z2 = x1x2’ + x3x4’ + x1’x2x4 z3 = x3x4’ + x2x3 + x2x4’ + x2’x3’x4 + x1x4’ z4 = x2’x3’x4 + x2x3’x4’ + x1x2’x3’ + x1x3x4 + x1’x2x4
NAND and NOR gate symbols
NAND/NOR Circuits Switching algebra: not directly applicable to NAND/NOR logic NAND and NOR gate symbols
Analysis of NAND/NOR Networks
Example: circles (inversions) at both ends of a line cancel each other
Synthesis of NAND/NOR Networks
Example: Realize T = w(y+z) + xy’z’
Full adder: performs binary addition of three binary digits Inputs: arguments A and B and carry-in C Outputs: sum S and carry-out C0 Example: Truth table, block diagram and expressions: S = A’B’C + A’BC’ + AB’C’ + ABC = A B C C0 = A’BC + ABC’ + AB’C + ABC = AB + AC + BC
Cf: forced carry C0(n-1): overflow carry Si = Ai Bi Ci C0i = AiBi + AiCi + BiCi Time required: Time per full adder: 2 units Time for ripple-carry adder: 2n units
Carry-lookahead adder: several stages simultaneously examined and their carries generated in parallel Generate signal Di = AiBi Propagate signal Ti = Ai Bi Thus, C0i = Di + TiCi To generate carries in parallel: convert recursive form to nonrecursive C0i = Di + TiCi Ci = C0(i-1) C0i = Di + Ti(Di-1 + Ti-1Ci-1) = Di + TiDi-1 + TiTi-1(Di-2 + Ti-2Ci-2) = Di + TiDi-1 + TiTi-1Di-2 + TiTi-1Ti-2Ci-2 …….. C0i = Di + TiDi-1 + TiTi-1Di-2 + … + TiTi-1Ti-2…T0Cf Thus, C0i = 1 if it has been generated in the ith stage or originated in a preceding stage and propagated to all subsequent stages
Implementation of lookahead for the complete adder impractical: Divide the n stages into groups Full carry lookahead within group Ripple carry between groups Example: Three-digit adder group with full carry lookahead Time taken: 4 time units for Cg1 Only 2 time units for Cg2 and other group carries
30-bit Adder Example: divide n stages into groups of three stages
Time taken: 4 + 2n/3 time units 50% additional hardware for a threefold speedup
Metal-oxide Semiconductor (MOS) Transistors and Gates
Complementary metal-oxide semiconductor (CMOS): currently the dominant technology Two types of transistors: nMOS and pMOS
Transmission Function of a Network
CMOS inverter and its transmission functions:
CMOS NAND/NOR Gates
Analysis of Series-parallel Networks
Algebra of MOS networks: isomorphic to switching algebra Example: Find the transmission function of the network and its complementary switch based and complex gate CMOS implementations Complementary switch based Complex gate
Analysis of Non-series-parallel Networks
Obtaining the transmission function: Tie sets: minimal paths between two terminals Cut sets: minimal sets of branches, when open, ensure no transmission between the two terminals
Synthesis of MOS Networks
Sneak paths in non-series-parallel networks: undesired paths that may change the transmission function Occur because of bilateral nature of MOS transistors Example: Design a minimal network with BCD inputs that produces a 1 whenever the input is 3 or a multiple of 3 Sneak path: z’xx’w – OK since it has no effect on the transmission function
Synthesis of MOS Networks (Contd.)
Example: Design a minimal network to realize T(w,x,y,z) = (0,3,13,14,15)
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# Question
Hi all
Need help with this question..thanks!
Wei Chong bought 15 identical stools and 7 identical desks for \$153.
2 desks cost as much as 3 stools. Find the cost of a stool?
This is a Quantity Each Product question.
2 desk = 3 stool
Stool = 2u
Desk = 3u
Quantity Each Product Stool 15 2u 30u Desk 7 3u 21u Total 51u
51u = 153
1u = 153 ÷ 51 = 3
3u = 3 × 3 = \$9
2 Replies 0 Likes ✔Accepted Answer
Amendment
Stool is 2u
2u = 3 ×2 = \$6
0 Replies 0 Likes
:This is a Quantity Each Product question.
2 desk = 3 stool
Stool = 2u
Desk = 3u
Quantity
Each
Product
Stool
15
2u
30u
Desk
7
3u
21u
Total
51u
51u = 153
1u = 153 ÷ 51 = 3
3u = 3 × 3 = \$9″
9 x 15 + (7/2) x 9 x 2 = 198 (not 153)
0 Replies 0 Likes
15 stools + 7 desks –> 153
3 stools –> 2 desks
15 stools –> 10 desks
10 desks + 7 desks –> 153
1 desk –> 153 ÷ 17
= 9
3 stools –> 2 × 9
= 18
Cost of 1 stool = \$18÷3
= \$6 (ans)
0 Replies 0 Likes
15 stools + 7 desks ——- 153
2 desks ——- 3 stools
7 desks ——- (3/2) x 7 = 10 1/2
10 1/2 + 15 = 25 1/2 stools ——- 153
1 stool ——- 6
Ans : \$6.
6 x 15 + (7/2) x 6 x 3 = 153
0 Replies 0 Likes
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Selina solutions
Grade 7
# Question 1
Q1) Multiply:
(i) 3x,\ 5x^2y\ and\ 2y
(ii) 5,\ 3a\ and\ 2ab^2
(iii) (iii) 5x + 2y and 3xy
(iv) 6a - 5b and - 2a
(v) 4a + 5b and 4a - 5b (vi) 9xy+2y^2\ and\ 2x-3y
(vii) -3m^2n\ +5mn-4mn^2and\ 6m^2n
(viii) 6xy^2-7x^2y^2+10x^3and\ -3x^2y^3
Solution 1:
(i) 3x,\ 5x^2y\ and\ 2y
= 3x\times5x^2y\times2y
= 15x^3y\times2y
= 30x^3y^2
(ii) 5,\ 3a\ and\ 2ab^2
= 5\times3a\times2ab^2
= 15a\times2ab^2
= 30a^2b^2
(iii) 5x + 2y and 3xy
= 3xy × (5x + 2y)
= 15x^2y+6xy^2
(iv) 6a - 5b and - 2a
= -2a × (6a - 5b)
= -12a^2+10ab
(v) 4a + 5b and 4a - 5b
(vi) 9xy+2y^2\ and\ 2x-3y
(vii) -3m^2n\ +5mn-4mn^2and\ 6m^2n
=6m^2n\left(-3m^2n+5mn-4mn^2\right)
= 18m^4n^2+30m^3n^2-24m^3n^3
(viii) 6xy^2-7x^2y^2+10x^3and\ -3x^2y^3
= 3x^2y^3\left(6xy^2-7x^2y^2+10x^3\right)
= -18x^3y^5+21x^4y^5-30x^5y^3
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0
# What two numbers multiply to 75 and add up to -3?
Updated: 9/23/2023
Wiki User
11y ago
72
Christian Sawayn
Lvl 10
3y ago
Wiki User
11y ago
The complex conjugate pair -1.5±
8.5294i, approximately, where i is the imaginary square root of -1.
Earn +20 pts
Q: What two numbers multiply to 75 and add up to -3?
Submit
Still have questions?
Related questions
-97
1209
### What two multiply numbers equal the sum of 100?
It's unclear whether you want two numbers that add to a sum or multiply to a product. 25 x 4 = 100 25 + 75 = 100
### what two numbers multiply to 75 and add to -20?
-5 and -15 because -5 + -15 equals -5 - 15 which is -20 and -5 multiplied by -15 = 75 (two negatives multiply to make a positive)
3 and 25
### What 2 numbers add to -27 and multiply to -75?
Try: 2.539014933 and -29.539014933
### What two prime numbers gives you 75 and when subtracted gives you 19?
None of this works. There aren't two prime numbers that multiply to 75 and the only two that add to it are 73 and 2. There aren't two prime numbers whose difference is 19. Two numbers that satisfy those conditions are 47 and 28, but 28 isn't prime.
2 x 75 = 150
### What two numbers can you multiply to get75?
How about: 3*25 = 75 as an example
### What two numbers multiply to get 75?
1 x 75, 3 x 25, 5 x 15
### What are two numbers you multiply to get 75?
1 x 75, 3 x 25, 5 x 15.
### What two numbers do you multiply to get 75?
1 x 75, 3 x 25, 5 x 15.
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There is a particularly neat way to derive Schrödinger’s equation, and to justify the “canonical substitution” rules for replacing energy and momentum with corresponding operators when we “quantize” an equation.
Take a particle in a potential. Its energy is given by
$$E=\frac{{\bf p}^2}{2m}+V({\bf x}),$$
or
$$E-\frac{{\bf p}^2}{2m}-V({\bf x})=0.$$
Now multiply both sides this equation by the formula $$e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}$$. We note that this exponential expression cannot ever be zero if the part in the exponent that’s in parentheses is real:
$$\left[E-\frac{{\bf p}^2}{2m}-V({\bf x})\right]e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}=0.$$
So far so good. But now note that
$$Ee^{i({\bf p}\cdot{\bf x}-Et)/\hbar}=i\hbar\frac{\partial}{\partial t}e^{i({\bf p}\cdot{\bf x}-Et)/\hbar},$$
and similarly,
$${\bf p}^2e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}=-\hbar^2{\boldsymbol\nabla}e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}.$$
This allows us to rewrite the previous equation as
$$\left[i\hbar\frac{\partial}{\partial t}+\hbar^2\frac{{\boldsymbol\nabla}^2}{2m}-V({\bf x})\right]e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}=0.$$
Or, writing $$\Psi=e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}$$ and rearranging:
$$i\hbar\frac{\partial}{\partial t}\Psi=-\hbar^2\frac{{\boldsymbol\nabla}^2}{2m}\Psi+V({\bf x})\Psi,$$
which is the good old Schrödinger equation.
The method works for an arbitrary, generic Hamiltonian, too. Given
$$H({\bf p})=E,$$
we can write
$$\left[E-H({\bf p})\right]e^{i({\bf p}\cdot{\bf x}-Et)/\hbar}=0,$$
which is equivalent to
$$\left[i\hbar\frac{\partial}{\partial t}-H(-i\hbar{\boldsymbol\nabla})\right]\Psi=0.$$
So if this equation is identically satisfied for a classical system with Hamiltonian $$H$$, what’s the big deal about quantum mechanics? Well… a classical system satisfies $$E-H({\bf p})=0$$, where $$E$$ and $${\bf p}$$ are eigenvalues of the differential operators $$i\hbar\partial/\partial t$$ and $$-i\hbar{\boldsymbol\nabla}$$, respectively. Schrödinger’s equation, on the other hand, remains valid in the general case, not just for the eigenvalues.
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# How We Can Impact Maths Learning & Teaching?
In the realm of mathematics education, innovative tools and methods play a crucial role in enhancing the learning experience for students. One such powerful tool that has proven to be instrumental in understanding trigonometry is the Unit Circle Chart. In this comprehensive guide, we will explore how the use of Unit Circle Charts can impact both the learning and teaching of mathematics.
## What Is A Unit Circle Chart?
A Unit Circle Chart is a visual representation of the relationships between angles and the trigonometric functions—sine, cosine, and tangent. It is a fundamental tool in trigonometry that provides a concise and organized way to grasp complex concepts.
### Components of a Unit Circle Chart
The Unit Circle Chart typically displays angles in both radians and degrees, allowing students to seamlessly transition between the two measurement systems.
Trigonometric Functions
The chart includes values for sine, cosine, and tangent corresponding to different angles, offering a quick reference for solving trigonometric equations.
## How Unit Circle Charts Enhance Mathematical Understanding?
Visual Learning Aids
The visual nature of Unit Circle Charts makes them an excellent aid for visual learners. By representing abstract mathematical concepts in a graphical format, students can better comprehend the relationships between angles and trigonometric functions.
Memorization and Retention
Sin Cos Tan Unit Circle Chart: A Mnemonic Device
The Sin Cos Tan Unit Circle Chart serves as a mnemonic device, aiding students in memorizing the values of sine, cosine, and tangent for common angles. This mnemonic strategy enhances long-term retention.
Trigonometry Unit Circle Chart: Simplifying Complex Concepts
The Trigonometry Unit Circle Chart simplifies the understanding of complex trigonometric concepts, such as the periodic nature of sine and cosine functions.
## Integrating Unit Circle Charts Into Mathematics Instruction
Interactive Learning Activities
Utilizing Technology: Interactive Unit Circle Apps
Teachers can incorporate technology, such as interactive Unit Circle apps, into their lessons to engage students actively. These tools allow students to explore the Unit Circle dynamically.
Classroom Demonstrations: Bringing the Unit Circle to Life
Engaging classroom demonstrations with physical Unit Circle charts help make abstract concepts more tangible, fostering a deeper understanding among students.
Incorporating Unit Circle Charts in Lesson Plans
Unit Circle Chart in Radians: A Seamless Transition
Integrating the Unit Circle Chart in radians into lesson plans ensures that students become proficient in both radians and degrees, a crucial skill for advanced mathematics.
Unit Circle Chart Values: Emphasizing Practical Application
Teachers can emphasize the practical application of Unit Circle Chart values in solving real-world problems, linking mathematical concepts to everyday scenarios.
## What Is The Significance Of The Unit Circle In Trigonometry?
The Unit Circle serves as a fundamental tool in trigonometry, providing a visual representation of the relationships between angles and trigonometric functions. It simplifies complex concepts and aids in problem-solving.
## How Does The Sin Cos Tan Unit Circle Chart Help With Memorization?
The Sin Cos Tan Unit Circle Chart acts as a mnemonic device, offering a systematic way to memorize the values of sine, cosine, and tangent for common angles. This aids in quick recall during mathematical problem-solving.
## Are There Online Resources For Interactive Unit Circle Learning?
Yes, several online platforms offer interactive Unit Circle tools and apps that allow students to explore and interact with the chart dynamically, enhancing their understanding of trigonometry.
## Can The Unit Circle Chart Be Used In Advanced Mathematics Courses?
Yes, The Unit Circle Chart is a versatile tool that finds applications in advanced mathematics courses, especially in fields like calculus and physics. Its principles remain foundational in higher-level studies.
## Conclusion: Empowering Mathematics Education with Unit Circle Charts
In conclusion, the integration of Unit Circle Charts into mathematics learning and teaching has a profound impact on student understanding and engagement. By leveraging the visual and mnemonic aspects of these charts, educators can transform complex trigonometric concepts into accessible and memorable lessons. As we continue to explore innovative tools in mathematics education, the Unit Circle Chart stands out as a timeless and indispensable resource.
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# Symmetry group
## Homework Statement
Show that symmetry operations for en greek vase build up a symmetry group.
## Homework Equations
For en greek vase we have
$$\Gamma=[e, C_{2},\sigma, \sigma^{'}]$$
And there are 3 conditions which must be fullfilled so that the elements will create a symmetry group $$1) (a\cdot b)\cdot c= a\cdot (b\cdot c)$$
$$2) a\cdot e= a$$
$$3) a\cdot a^{-1}=e$$
## The Attempt at a Solution
So we know that the vase is invariant under $$180^{0}$$ so it is of $$C_{2} type$$
do I understand correctly$$C_{2}\cdot C^{-1}_{2}=e$$ rotation $$180^{0}$$ and another one $$180^{0}$$ in the opposite direction
second condition-($$a\cdot e=a$$)
can we write then $$C_{2}\cdot e=C_{2}$$???
How will it work for the condition 1?$$(a\cdot b)\cdot c=a(b\cdot c)$$
Can we show it in this way? $$(C_{2}\cdot e)\cdot C^{-1}_{2}=C_{2}\cdot (e\cdot C^{-1}_{2})\rightarrow e=e$$
How can we show it with using other symmetry elements? $$[\sigma, \sigma{'}, e]$$
for example$$(C_{2}\cdot \sigma)\cdot e=C_{2}\cdot(\sigma\cdot e)$$???
## The Attempt at a Solution
fzero
Homework Helper
Gold Member
There's actually a condition that you must verify before considering the ones that you've listed. It is closure, that the product of two elements of $$\Gamma$$ is also an element belonging to $$\Gamma$$. Therefore, the first thing you want to do is build the multiplication table for the elements of $$\Gamma$$.
When you know the rules for multiplication, verifying associativity will be fairly simple.
you mean I have to calculate
$$C_{2}\cdot e=$$
$$C_{2}\cdot C_{2}=$$
$$C_{2}\cdot \sigma=$$
$$C_{2}\cdot \sigma^{'}=$$
$$\sigma^{'}\cdot \sigma{'}=$$
$$e\cdot e=$$
$$e\cdot C_{2}=$$
$$e\cdot \sigma=$$
$$e\cdot \sigma^{'}=$$
and so on?
I thought that these 3 conditions had to be fullfilled to call these elements as a symmetry group
Last edited:
fzero
Homework Helper
Gold Member
Like I said, the requirement that the product of two elements of a set is another element in the set is also a requirement to have a group. When you write
$$(a\cdot b)\cdot c= a\cdot (b\cdot c),$$
you're assuming that $$(a\cdot b)\cdot c$$ is actually in $$\Gamma$$.
However, it's not just that you have to verify closure. It's also that knowing the multiplication table is necessary to verify condition 1 anyway. How could you say that
$$(C_2 \cdot \sigma)\cdot \sigma' = C_2 \cdot ( \sigma\cdot \sigma')$$
if you don't know what $$C_2 \cdot \sigma$$ is equal to?
yes you are right, thank you, but here I meet another problem. I do not know what I will get when for example
$$C_{2}\cdot \sigma=$$ or
$$C_{2}\cdot \sigma^{'}=$$
first I rotate the vase and then mirror reflection....
Last edited:
fzero
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# Statistics math problems | Mathematics homework help
Module 5 Homework Assignment
1. Find the critical value or values of χ2 based on the given information. H1: σ > 26.1, n = 9, α = 0.01
Solution:
2. Test the claim that for the adult population of one town, the mean annual salary is given by
μ = 30,000. Sample data are summarized as n = 17, = \$22,298, and s = \$14,200. Use a significance level of α = 0.05. State the null and alternative hypotheses.
Solution:
3. (Refer to Question 2) Find the test statistic.
Solution:
4. (Refer to Question 2) Find the critical value(s) and state and explain decision about null hypothesis.
Solution:
5. (Refer to Question 2) State the conclusion in non-technical terms.
Solution:
6. A manufacturer uses a new production method to produce steel rods. A random sample of 17 steel rods resulted in lengths with a standard deviation of 4.7 cm. At the 0.10 significance level, test the claim that the new production method has lengths with a standard deviation different from 3.5 cm, which was the standard deviation for the old method. State the null and alternative hypotheses.
Solution:
7. (Refer to Question 6) Find the test statistic.
Solution:
8. (Refer to Question 6) Find the critical value(s) and state and explain the decision about the null hypothesis.
Solution:
9. (Refer to Question 6) State conclusion in non-technical terms.
Solution:
10. (Refer to Question 6) Calculate and show the confidence interval based on the significance level given above. Are the results the same as in question 8 and 9?
Solution:
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Trang chủ / Toán Kangaroo / Lớp 7-8
### Đề thi toán Kangaroo Mỹ lớp 7-8 năm 2010
1. 3 points questions
How much is 12 + 23 + 34 + 45 + 56 + 67 + 78 + 89?
2. How many axes of symmetry does the figure have?
3. oy kangaroos are packed for shipment. Each of them is packed in a box,which is a cube. Exactly eight boxes are packed tightly in a bigger cubic cardboard box. How many kangaroo boxes are on the bottom floor of this big cube?
4. The perimeter of the figure is equal to...
5. Eleanor draws the six vertices of a regular hexagon and then connects some of the 6 points with lines to obtain a geometric figure. Then this figure is surely not a...
6. If we type seven consecutive integer numbers and the sum of the smallest three numbers is 33, what is the sum of the largest three numbers?
7. After stocking up firewood, the worker summed up that from acertain number of logs he made 72 logs with 53 cuts. He saws only one log at a time. How many logs were there at the beginning?
8. There are seven bars in the box. They are 3 cm × 1 cm in size. The box is of size 5 cm × 5 cm. Is it possible to slide the bars in the box so that there will be room for one more bar? At least how many bars must be moved to accomplish this?
9. A square is divided into 4 smaller equal-sized squares. All the smaller squares are coloured in dark and light grey. How many different ways are there to colourthe given square? (Two colourings are considered the same if one can be rotated to give the other, as shown)
10. The sum of the first hundred positive odd integers subtracted from the sum of the first hundred positive even integers is
11. 4 points questions
Grandma baked a cake for her grandchildren who will visitherin the afternoon. Unfortunately she forgot whether only 3, 5 or all 6 of her grandchildren will come over. She wants to ensure that every child gets the same amount of cake. Then, to be prepared for all three possibilities she better cut the cake into
12. Which of the following is the smallest two-digit number that is not the sum of three different one-digit numbers?
13. Cathy needs 18 min to make a long chain by connecting three short chains with extra chain links. How long does it take her to make a really long chain by connecting six short chains in the same way?
14. In quadrilateral ABCD we have AD= BC, DAC= 50º, DCA= 65º, ACB= 70º (see the fig.). Find the value of angle ABC
15. Andrea has wound some rope around a piece of wood. She rotates the wood as shown with the arrow.
Front side
What is the correct back side of the piece of wood?
Back side:
16. There are 50 bricks of white, blue and red colour inabox. The number of white bricks is eleven times the number of blue ones. There are fewer red ones than white ones, but more red ones than blue ones.How many fewer red bricks are there than white ones?
17. On the picture ABCDis a rectangle, PQRSis a square. The shaded area is half of the area of rectangle ABCD. What is the length of the PX?
18. What is the smallest number of straight lines needed to divide the plane into exactly 5 regions?
19. If a–1 = b+ 2 = c–3 = d+ 4 = e–5, then which of the numbers a, b, c, d, e is the largest?
20. The logo shown is made entirely from semicircular arcs of radius 2 cm, 4 cm or 8 cm. What fraction of the logo is shaded?
21. 5 points questions
In the figure there are nine regions inside the circles. Put all the numbers from 1 to 9 exactly one in each region so that the sum of the numbers inside each circle is 11.
Which number must be written in the region with the question mark?
22. At a barter market, the goods have to be exchanged according to the price list stated in the chart. At least how many hens doesMr. Blackhave to bring to the market, to be able to take away one goose, one turkey and one cock?
23. A paper strip was folded three times in half and then completely unfolded so that you can still see the 7 folds going up or down. Which of the following views from the side cannot be obtained in this way?
24. On each of 18 cards exactly one number is written, either 4 or 5. The sum of all numbers on the cards is divisible by 17. On how many cards is the number 4 written?
25. The natural numbers from 1 to 10 are written on the blackboard. The students in the class play the following game: a student deletes 2 of the numbers and instead of them writes on the blackboard their sum decreased by 1; after that another student deletes 2 of the numbers and instead of them writes on the blackboard their sum decreased by 1; and so on. The game continues until only one number remains on the blackboard. The last number is:
26. In a town there are only knights and liars. Every sentence spoken by a knight is true, every sentence spoken by a liar is false. One day some citizens were in a room and three of them spoke as follows:1) The first one said: «There are no more than three of us in the room. All of us are liars». 2) The second said: «There are no more than four of us in the room. Not all of us are liars». 3) The third said: «There are five of us in the room. Three of us are liars. How many people are in the room and how many liars are among them?
27. A Kangaroo has a large collection of small cubes 1 × 1 × 1. Each cube is a single colour. Kangaroo wants to use 27 small cubes to make a 3 × 3 × 3 cube so that any two cubes with at least one common vertex are of different colours. At least how many colours have to be used?
28. The biggest equilateral triangle consists of 36 smaller equilateral triangles with area 1 cm2 each. Find the area of ∆ABC
29. Five friends have each21, 32, 17, 11 and16 candies. They want to give some candies to another friend. What is the minimum number of candies that they can give so that the initial group of five friends could share equally the candies between them?
30. In the figure, α= 7° and the segments OA1,A1A2, A2A3, ... are all equal. The lengths OA form a numerically increasing sequence. What is the greatest number of segments that can be drawn in this way?
| 1,500 | 6,040 |
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# S.O.S. Mathematics CyberBoard
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Post subject: tricky questionPosted: Sat, 15 Jan 2011 23:57:49 UTC
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Joined: Sun, 9 Jan 2011 03:12:59 UTC
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i need help with this question:
jane is 20 years old today. jane is going to put \$1000 into her savings account on her 21th birthday and again on every birthday for 20 payments. she earn 5%, paid annually. how much money will be in the account after she collects her interest and makes her 20th payment?
this is my attempt:
i calculate the PV for 19 years first
pmt=1000, n=19, i=5, fv=0, pv=?
pv = 30539
then, calculate the principle 1000 x 19 = 19000
then take 30539- 19000= 11539 interest
so if jane collect interest 11539 on the 19 payment , the remaining in the account is 19000.
she again make last payment 1000 into account on her 40th birthday.
the account would left with 20,000
Am i doing this correct, can anyone give me some guidance.
greatly appreciate.
ann
[/i]
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Post subject: Posted: Sun, 16 Jan 2011 05:07:19 UTC
Member of the 'S.O.S. Math' Hall of Fame
Joined: Sun, 24 Jul 2005 20:12:39 UTC
Posts: 3690
Location: Ottawa Ontario
Ann, were you not given formulas by your teacher?
What you're doing is completely erroneous.
pmt = 1000 ; n = 20 ; i = .05 ; fv = ?
fv = 33,065.9541....
Or by formula:
fv = 1000(1.05^20 - 1) / .05 = 33,065.9541....
_________________
I'm not prejudiced...I hate everybody equally!
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Post subject: Posted: Mon, 28 Feb 2011 22:14:41 UTC
S.O.S. Newbie
Joined: Mon, 28 Feb 2011 17:01:43 UTC
Posts: 2
Denis is right. This is a trivial annuity-immediate question. You can google the concept "annuity-immediate." For example: http://en.wikipedia.org/wiki/Annuity_(finance_theory). The formula you need is
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Q:
# Your classmate is unsure about how to use side lengths to determine the type of triangle. How would you explain this to your classmate?
Accepted Solution
A:
1) Equilateral Triangles: These triangles have the same side length and same angle for all sides. Knowing this, if the side lengths given are all the same, the triangle will be classified as an equilateral triangle.
2) Isosceles Triangles: These triangles have two sides that have the same side length and one that is not the same. Given this, if the side lengths are all given, the triangle with two same sides are classified as isosceles triangle.
3) Scalene Triangles: These triangles have all sides associated with a different value. If all side lengths are given and they have different values, this triangle would classify as a scalene triangle.
Hope this helped :)
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# Formal proof
Last updated
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (called well-formed formulas in the case of a formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. [1] It differs from a natural language argument in that it is rigorous, unambiguous and mechanically verifiable. [2] If the set of assumptions is empty, then the last sentence in a formal proof is called a theorem of the formal system. The notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concepts of Fitch-style proof, sequent calculus and natural deduction are generalizations of the concept of proof. [3] [4]
## Contents
The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formed formula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus (of some formal system) to the previous well-formed formulas in the proof sequence.
Formal proofs often are constructed with the help of computers in interactive theorem proving (e.g., through the use of proof checker and automated theorem prover). [5] Significantly, these proofs can be checked automatically, also by computer. Checking formal proofs is usually simple, while the problem of finding proofs (automated theorem proving) is usually computationally intractable and/or only semi-decidable, depending upon the formal system in use.
## Background
### Formal language
A formal language is a set of finite sequences of symbols. Such a language can be defined without reference to any meanings of any of its expressions; it can exist before any interpretation is assigned to it that is, before it has any meaning. Formal proofs are expressed in some formal languages.
### Formal grammar
A formal grammar (also called formation rules) is a precise description of the well-formed formulas of a formal language. It is synonymous with the set of strings over the alphabet of the formal language which constitute well formed formulas. However, it does not describe their semantics (i.e. what they mean).
### Formal systems
A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions.
### Interpretations
An interpretation of a formal system is the assignment of meanings to the symbols, and truth values to the sentences of a formal system. The study of interpretations is called formal semantics. Giving an interpretation is synonymous with constructing a model.
## Related Research Articles
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.
In logic, more precisely in deductive reasoning, an argument is sound if it is both valid in form and its premises are true. Soundness also has a related meaning in mathematical logic, wherein logical systems are sound if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system.
In mathematics and logic, a theorem is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems. A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system. As a result, the proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.
Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F", requiring only a few apparently innocuous logical deduction rules. Since F is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic.
Proof theory is a major branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of the logical system. As such, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
A rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion. For example, the rule of inference called modus ponens takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic, in the sense that if the premises are true, then so is the conclusion.
In mathematical logic, sequent calculus is, in essence, a style of formal logical argumentation where every line of a proof is a conditional tautology instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than David Hilbert's earlier style of formal logic where every line was an unconditional tautology. There may be more subtle distinctions to be made; for example, there may be non-logical axioms upon which all propositions are implicitly dependent. Then sequents signify conditional theorems in a first-order language rather than conditional tautologies.
In mathematical logic, a sequent is a very general kind of conditional assertion.
In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
Metalogic is the study of the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.
A formal system is used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system".
An object language is a language which is the "object" of study in various fields including logic, linguistics, mathematics, and theoretical computer science. The language being used to talk about an object language is called a metalanguage. An object language may be a formal or natural language.
In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning.
In mathematical logic, a theory is a set of sentences in a formal language. In most scenarios, a deductive system is first understood from context, after which an element of a theory is then called a theorem of the theory. In many deductive systems there is usually a subset that is called "the set of axioms" of the theory , in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
In mathematical logic, formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. These rules only address the location and manipulation of the strings of the language. It does not describe anything else about a language, such as its semantics. .
Logic is the systematic study of valid rules of inference, i.e. the relations that lead to the acceptance of one proposition on the basis of a set of other propositions (premises). More broadly, logic is the analysis and appraisal of arguments.
## References
1. "The Definitive Glossary of Higher Mathematical Jargon — Rigor". Math Vault. 2019-08-01. Retrieved 2019-12-12.
2. Kassios, Yannis (February 20, 2009). "Formal Proof" (PDF). cs.utoronto.ca. Retrieved 2019-12-12.
3. The Cambridge Dictionary of Philosophy, deduction
4. Barwise, Jon; Etchemendy, John Etchemendy (1999). Language, Proof and Logic (1st ed.). Seven Bridges Press and CSLI.
5. Harrison, John (December 2008). "Formal Proof—Theory and Practice" (PDF). ams.org. Retrieved 2019-12-12.
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English
# Poisson Distribution
## Poisson Distribution Tutorial
##### Definition:
In statistics, poisson distribution is one of the discrete probability distribution. This distribution is used for calculating the possibilities for an event with the given average rate of value(λ). A poisson random variable(x) refers to the number of success in a poisson experiment.
#### Formula:
f(x) = eλx / x!
###### where,
λ is an average rate of value. x is a poisson random variable. e is the base of logarithm(e=2.718).
##### Example:
Consider, in an office 2 customers arrived today. Calculate the possibilities for exactly 3 customers to be arrived on tomorrow.
###### Step 1:
Find e. where, λ=2 and e=2.718 e = (2.718)-2 = 0.135.
###### Step 2:
Find λx. where, λ=2 and x=3. λx = 23 = 8.
###### Step 3:
Find f(x). f(x) = eλx / x! f(3) = (0.135)(8) / 3! = 0.18. Hence there are 18% possibilities for 3 customers to be arrived on tomorrow.
This tutorial will guide you to calculate the poisson distribution.
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# Volume
What is volume? It is a word that can have several meanings, such as a number in a series of books; but in terms of measurement we are looking at how much space an object or liquid occupies. For example, if I have an empty bottle and want to fill it up, how much liquid can fit in there?
Volume is measured using litres and millilitres, but for larger objects it can be measured in cubic metres. An alternative to millilitres is cubic centimetres, in that they are the same size, just different names, both measuring small volumes. Let us see how this looks below:
If we take a small cube, which has each side measuring 1 centimetre in length, we have a volume of 1 cubic centimetre.
If we have something similar but bigger, so that each side is 10 centimetres in length, we see, from the image below, that it has a volume of 1000 cubic centimetres, which we also call 1 litre:
But not everything is a nice perfect cube shape, therefore volumes are not always easy to measure for solid objects, but usually we only want to know the volume of something liquid. We can use litres and cubic centimetres, or even cubic metres, as well as millilitres, for measuring solid objects, or space that something might fit into, e.g. the space of your car boot, as well as for liquids such as drinks, water, and for gas.
For easy reference:
## 1000 litres = 1 cubic metre
Here are some examples:
Item Volume Bottle of water bought from a supermarket Small: 500 ml medium: 1 litre large: 2 litres Luggage space in car boot Small car: 200 litres Medium car: 380 litres Large car: 500 litres Estate car: 540 litres People carrier: with all seats upright, 330 litres with only 2 rows of seats, 800 litres with 1 row of seats, luggage piled up to the roof, 2.6 cubic metres (2600 litres) Fridge internal capacity Small: 100 litres Medium: 150 litres Large: 350 litres Wine Small glass: 150 millilitres Larger glass: 250 ml Bottle: 75 cl (750 ml) Glasses Small glass: 220 ml Pint glass: 568 ml Tea Tea mug: 250 ml Tea pot: 1.1 L = 1100 ml Milk carton Carton of milk, e.g. soya milk: 1 litre Kettle Max usable capacity: 1.7 litre Bath 225 litres Swimming pool If we take large sized pool, length 50 metres, width 25 m, shallow end depth of 2 m, deep end depth of 3 m, then the volume is 3125 cubic metres, which equates to 3 125 000 litres of water. A smaller pool, such as pictured, with length 18 m, width 6 m, depths 1 m to 2 m, has a volume of 162 m3 which is 162 000 litres.
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# Good examples of proof by contradiction?
In later courses on automata theory, many students just seem incapable of getting a proof that a language isn't regular right, be it using the pumping lemma (see also the many questions on the matter on http://math.stackexchange.com) or the (often easier) use of closure properties.
As I am also teaching a first course in discrete matematics, were we go over proof techniques, I have tried (rather unsuccessfully, regrettably) to get the idea across. I would like to have more easy to grasp examples where the technique is natural (and needed, contrived examples just lead to "the prof is off his rocker again, this is trivial to do by ...", and so are more a distraction than a help).
• How about proving that the square root of 2 is an irrational number? Commented Mar 16, 2014 at 15:43
• To add on to @WeirdstressFunction's comment, I don't know of a way to prove that $\sqrt 2$ is irrational that isn't a proof by contradiction.
– user37
Commented Mar 16, 2014 at 16:07
• My discrete math professor proved If 3n+2 is odd, then n is odd by contradiction as an example. It's also able to be done using contrapositive, but she highlighted the differences by proving both ways. Commented Mar 17, 2014 at 4:08
• @Mike, $\sqrt2$ is a root of $x^2-2$. Any rational root of this must have denominator dividing $1$ and numerator dividing $-2$, and so must be $\pm 1$ or $\pm 2$. None of these square to $2$, so none are $\sqrt2$ and hence $\sqrt2$ is not rational. Commented Mar 17, 2014 at 4:14
• @Mike The square of a fraction in lowest terms is again in lowest terms, so a rational number is a perfect square if and only if its numerator and denominator are square numbers. Commented Mar 18, 2014 at 4:48
Here are some good examples of proof by contradiction:
1. Euclid's proof of the infinitude of the primes. (Edit: There are some issues with this example, both historical and pedagogical. See Mike F.'s answer and the ensuing discussion.)
2. The famous proof that $$\sqrt{2}$$ is irrational. (I don't particularly like this one---there are better ways of proving this. See my comment above.)
3. The sum of a rational number and an irrational number is irrational.
4. Cantor's diagonal argument that $$\mathbb{R}$$ is uncountable is a proof by contradiction. (Edit: As Santiago Canez points out in the comments, this example and the next are perhaps better stated as direct proofs.)
5. Similarly, there's the proof that there is no bijection from a set $$X$$ to the power set of $$X$$.
6. Russell's proof that there exists no set of all sets.
7. The proof of Gödel's incompleteness theorem.
8. This Math Stack Exchange post has a nice simple proof that $$\displaystyle\sum_{k=1}^n \frac1k$$ is never an integer for $$n\geq 2$$.
9. There are lots of basic statements about Diophantine equations that can be proven by contradiction. For example, the statement "the equation $$4x^2-y^2 = 1$$ has no integer solutions for $$x$$ and $$y$$" has a simple contradiction proof. (Factor the left side.)
10. This Math Stack Exchange post has a simple proof that there exist infinitely many primes $$p$$ such that $$p+2$$ is not prime.
11. The proof of Eisenstein's criterion for irreducible polynomials is a proof by contradiction. There are many simpler statements along these lines (e.g. proving that specific polynomials have no integer roots) that can be proved by contradiction.
12. This Math Stack Exchange post has a simple proof using the trace that if $$A$$ and $$B$$ are $$n\times n$$ matrices and $$AB-BA = B$$, then $$B$$ cannot be invertible.
13. There's a simple proof by contradiction that there does not exist a continuous function $$f\colon\mathbb{R}\to\mathbb{R}$$ so that $$f(f(x))=-x$$. (It must be a bijection, so it's either increasing or decreasing, so . . .)
14. You can prove by contradiction that there's no embedding of the complete graph $$K_5$$ in the plane using Euler's formula.
15. The solution to the Seven Bridges of Königsberg problem is essentially a proof by contradiction.
16. Twenty five boys and twenty five girls sit around a table. Prove that it is always possible to find a person both of whose neighbors are girls.
• Cantor's diagonal argument is NOT a proof by contradiction, it is a direct proof that no function from $\mathbb N$ to $\mathbb R$ is surjective. Similarly, your fifth example is actually a direct proof that no function from a set to its power set is surjective. Commented Mar 17, 2014 at 4:16
• @SantiagoCanez I suppose you can phrase the proof that way, but it can also be phrased as a contradiction proof. In particular, you can certainly find lots of books written by perfectly good mathematicians in which the proof is described as a proof by contradiction. As with Matt F.'s comment below, I guess I don't see the point of purposely avoiding the contradiction argument. Is there some reason that contradiction proofs should be avoided at all costs? Commented Mar 17, 2014 at 5:09
• Overuse of proof by contradiction leads students to believe that every proof should be a proof by contradiction, meaning that it becomes the first strategy they attempt eve though most of the time it makes things more confusing. I'm not saying that contradiction couldn't be used here, but that contradiction should only be used when it is necessary so that students develop better intuition as to how to approach proof writing. Commented Mar 17, 2014 at 12:52
• I think that Cantor's argument really is a proof by contradiction. It is true that it brilliantly "constructs" an element not in the image of any given map $f: S \mapsto 2^S$...but the argument for that is by contradiction. What else? Commented Mar 18, 2014 at 5:25
• The traditional proof that $\sqrt2$ is irrational is not a good first introduction to proof by contradiction, in my experience with students. Firstly, the thing you contradict is usually introduced by the writer "in lowest terms". This feels like cheating to many students. Secondly, there is a bit of number theory ($a^2$ is a multiple of 2 means that $a$ is, which is actually because 2 is prime) and most students (at least most of mine) have no experience in that. Commented Aug 3, 2014 at 20:08
Many of Jim Belk's answers are good. But let me state for the record, because it always comes up:
Euclid's proof of the infinitude of primes is not a proof by contradiction.
Look at Euclid's text, e.g. the translation here: "Prime numbers are more than any assigned multitude of prime numbers." The proof is constructive.
I have found that proving this theorem by contradiction confuses students. I would encourage others to prove it constructively instead. In any case: do not attribute the proof by contradiction to Euclid.
• It is true that Euclid himself phrased the proof in a "direct" way, but only by stating the theorem as "every finite set of primes is not equal to the set of all primes". If you use Euclid's proof to prove the statement "the set of primes is infinite", then you are using a proof by contradiction. (Continued in the next comment.) Commented Mar 17, 2014 at 4:54
• Moreover, I don't think this distinction is all that important: you can make almost any contradiction proof into a direct proof by changing the statement of the theorem. For example, it's easy to give a direct proof of the theorem "the square of any rational number is not equal to two", but what of it? Commented Mar 17, 2014 at 4:55
• @Jim Belk: Euclid's proof gives a certain algorithm for, given any set of $N$ prime numbers, producing a prime number which is not in the set. Not every proof can be turned into an algorithm, and it is important to know which can. Now in fact any proof, no matter how indirect, of the infinitude of primes, leads to an algorithm for producing primes (namely trial factorization) but Euclid's proof gives an explicit upper bound on the size of the $n$th prime. This is the beginning of analytic number theory. Commented Mar 18, 2014 at 3:37
• @PeteL.Clark That is a good point. Euclid's original proof is in some sense much more algorithmic that the typical indirect version that is given, and indeed phrasing it as a proof by contradiction obscures its effective nature, and leaves students without an algorithm for generating an infinite list of primes. Commented Mar 18, 2014 at 4:30
• @PeteL.Clark your link does not seem to work anymore. Can you update? Commented Jan 10, 2017 at 8:29
I would like to use this as an opportunity to make an important distinction.
Proof by contradiction is an argument of the form:
Assume $$\neg p$$
Argue a contradiction under this assumption.
Conclude $$p$$.
Proof of negation is an argument of the form
Assume $$p$$
Argue a contradiction under this assumption.
Conclude $$\neg p$$
I learned about the difference from Andrej Bauer here:
A classical mathematician might not distinguish these proofs, because they think $$\neg (\neg p) \equiv p$$, but a constructive mathematician will make this distinction.
I would like to go through Jim Belk's list to illustrate the difference:
1. Infinitude of primes. As noted, this can be phrased as a proof by contradiction, but it can also be viewed as a completely constructive result: given a list of prime numbers, it gives an algorithm for constructing a new prime which is not in the list.
2. Irrationality of $$\sqrt{2}$$. This is proof of negation. The definition of "irrational" is "not rational". You prove the negation of "$$\sqrt{2}$$ is rational" by assuming it is and obtaining a contradiction.
3. Sum of a rational and irrational is irrational. Again, this is proof of negation.
4. Cantor's diagonalization argument: To prove there is no bijection, you assume there is one and obtain a contradiction. This is proof of negation, not proof by contradiction. I will point out that, similar to the infinitude of primes example, this can be rephrased more constructively. Given an injection $$\mathbb{N} \to \mathbb{R}$$, this argument explicitly produces an element of $$\mathbb{R}$$ which is not in the image.
5. No bijection from $$X$$ to $$\mathcal{P}(X)$$: similar remarks to Cantor's argument apply. As usually phrased it is proof of negation, and it can be rephrased more constructively as a recipe which takes an injection from $$X$$ to $$\mathcal{P}(X)$$ and produces an element of $$\mathcal{P}(x)$$ which is not in the image.
6. Russel's proof that there is no set of all sets. I am not sure this counts as a formal argument. It is more of a metamathematical theorem, that we should not attempt to construct a formal system which allows unbounded set formation.
7. $$\vdots$$
I think all of the other examples are also proof of negations, rather than proof by contradiction. Many of the theorems can be rephrased more powerfully without using negation in the theorem statement at all, and they have direct proofs in this case. None of them, that I can see, can both be rephrased without negation in the statement and require proof by contradiction, rather than direct proof, to demonstrate them.
Usually proof by contradiction can be avoided, and doing so creates direct proofs which are more constructive. Even for the non-constructive mathematician this is good mathematical hygiene: the intermediate results proven during a proof by contradiction are useless to your later work (they only hold under a false premise), while the intermediate results obtained in the direct proof are all immediately useful in real circumstances.
Proof of negation is unavoidable though. We can actually define $$\neg p$$ as $$p \implies F$$. If you are trying to prove that a certain number is not rational, we must show that assuming it is rational leads to contradiction.
## Example 1
Question:
Prove that there is only one circle with $AB$ as its diameter.
Assumption:
Assume that there are 3 circles $C_1$, $C_2$, and $C_3$ passing through the points $A$ and $B$. $C_1$ and $C_2$ are concentric and $C_1$ and $C_3$ are not concentric. $C_1$ and $C_2$ have different radii and $C_3$ has any radius. Let $C_1$ be on the midpoint of $AB$ such that $AB$ is its diameter.
• As $C_1$ and $C_2$ have different radii, points $A$ and $B$ cannot be on the circle $C_2$.
• As $C_3$ is not on the middle of $AB$, $AB$ cannot be its diameter.
Conclusion:
So there is only one circle $C_1$ with $AB$ as its diameter.
## Example 2
Question:
Prove that $\sqrt{2}$ is an irrational number.
Assumption:
Let $\sqrt 2$ be a rational number. So it can be represented as $\sqrt{2}=\frac{m}{n}$ where $m$ and $n$ are natural numbers without common factors other than $1$.
Squaring both sides, we get \begin{align} 2 &=\frac{m^2}{n^2}\\ m^2 &= 2n^2 \end{align}
Because $m^2$ is a multiple of $2$ then $m^2$ is an even number. Recall that
The square of an even number is even.
it implies that $m$ is also even. Let $m=2k$ where $k$ is any natural number. Substituting it for $m$, we get \begin{align} (2k)^2 &=2n^2\\ 4k^2 &= 2 n^2\\ n^2 &= 2k^2 \end{align}
With the same reasoning, $n$ is even. As both $m$ and $n$ are even numbers, 2 becomes their common factors so it contradicts the assumption that they have no common factors other than 1.
Conclusion:
$\sqrt 2$ cannot be represented as a ratio of two natural numbers without common factor other than 1. It implies that $\sqrt 2$ is irrational.
• "Recall that 'The square of an even number is even.' It implies that m is also even." Sorry, as a number theorist I have to frown at this. Try it with e.g. "a number divisible by 4" instead of "an even number". (Also, not to pick, but: it seems to me that there is only one circle with a given diameter is geometrically obvious and easier to verbally nail down in other ways: if you have the diameter then the midpoint is the center and half the length of the diameter is the radius. So you know the circle.) Commented Mar 18, 2014 at 5:01
• @Pete L. Clark: reply to an old comment, to be sure, but perhaps they meant the square of only an even integer is even, so $m$ can't be odd? Literally I agree because in some conceivable universe every perfect square might be even in which case one would draw no conclusion, making me think they misspoke a bit sloppily. Commented Sep 22, 2020 at 16:09
Something more trivial could be the following one: $$a^2=0\Rightarrow a=0.$$ Assume that $$a^2=0$$ and at the same time $$a\neq0$$ for some $$a\in\mathbb{R}.$$ Then, there exists $$a^{-1}\in\mathbb{R}$$ so: $$a^2=0\Rightarrow a^{-1}a^2=0\Rightarrow a=0,$$ so you indeed need both $$a^2=0$$ as well as $$a\neq0$$ to arrive to a contradiction.
Claim: The function $$f:\mathbb{R} \to \mathbb{R}$$ defined by $$f(x) = x^3+x-2$$ has one and only one root.
We can see that $$f(1) = 0$$ so $$f$$ has at least one root. Assume the the contrary that $$f$$ has an additional root $$z$$. Then, by the mean value theorem for derivatives, there is a $$c$$ between $$1$$ and $$z$$ so that $$f'(c) = 0$$. However, $$f'(c) = 3c^2 + 1 > 1$$. We have arrived at a contradiction, so the hypothesis that $$f$$ has more than $$1$$ root false. Thus $$f$$ has one and only one root.
Let $$x$$ be a root.
\begin{align} & \hphantom{fdsfsf} x^3 + x - 2 = 0\\ &\iff (x-1)(x^2+x+2) =0\\ &\iff (x-1)\left[(x+\frac{1}{2})^2 + \frac{7}{4}\right] = 0\\ &\iff x-1 = 0\\ &\iff x= 1 \end{align}
We can justify the equivalence of the third and fourth line by noting that $$\left[(x+\frac{1}{2})^2 + \frac{7}{4}\right] \geq \frac{7}{4}$$
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# The Beauty of Number Patterns
## Teaching your child about patterns can be a fun and motivating experience.
By Jacqueline Dwyer
Posted
The world around us provides virtually limitless ways to teach number patterns to children. I found that our local place of worship, the mall, the park, and our public swimming pool were all wonderful places to investigate number patterns. We visited each place so my son could copy the pattern of the floor tiles, mosaics, and stained glass windows. We found that using graph paper made it easier for him to draw the patterns he saw. Having these visual representations in front of us, made it easy to calculate the number patterns they each contained. Also, the repetition within the patterns gave us a great opportunity to practice multiplication tables. Since my son was essentially making scale drawings, this led to a discussion of architecture. We pondered how each place could be improved upon, in terms of space, seating, and lighting. We shared our ideas with our local homeschooling group. As a group, we created plans for improving some of the venues. Next, we sent a letter to our local government outlining our ideas and explaining the reasoning behind them. When we went to the park to draw patterns, we were also able to combine math and health. We measured and recorded our heartbeats at rest, then again after walking, hopping, and running. Later, at home, we made a graph of ascending and descending number patterns according to how vigorously we had exercised.
### Investigating Functions
In addition to the more concrete number patterns, we also looked at abstract, symbolic representations, such as input-output function machines, which are a stepping stone to algebra. These are not actual machines, of course, but are often drawn as computers where numbers go in and come out. They can be an enjoyable way to introduce children to relationships between numbers. Each machine has three main areas: input, rule, and output. Once the rule has been established, e.g. “+2“, the child inputs a number (e.g. “2“), follows the rule (“+2“), then calculates the output (“4“). Once your child is familiar with how function machines work, you can give him the input and output numbers, having him figure out the rule. Here is an example of a basic input-output function machine.
### Making Predictions Using Number Patterns
You can base input-output function machines on anything you like. Because we live near the ocean, and are subject to hurricanes, we collected data from our immediate environment. We looked online at predicted wave surge heights for different categories of hurricane. Next, we based our machine on average wave height (input), storm surge (rule), and post-hurricane wave height (output). We calculated input-rule-output for several nearby towns. We projected our forecast according to the strength of the hurricane and the distance and elevation of each town from the shore. While it makes sense that the further a town lies inland, the safer it is from the effects of a hurricane, it was interesting to confirm this hypothesis by calculating it mathematically.
Number Pattern lesson plans:
Visual and Number Patterns
Students describe, extend, and create their own patterns based on numbers, attributes, and geometry.
Explore Number Patterns using a Hundreds Chart
Students explore six different number patterns within a hundreds chart.
Patterns and Possibilities
Discover patterns using a variety of methods, including videos and hands-on activities.
Guess My Rule
Using patterns, functions, and Algebra, this lesson plan provides great ideas and worksheets for teaching and practicing math patterns. It can be adjusted for desired grade level
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# Find the equation of the tangent to the curve with exponential function
The question is as follows: Find the equation of the tangent to the curve $y = xe^{2x}$ at the point $(\frac{1}{2}, \frac{e}{2})$.
Now I figured out that $\frac{dy}{dx} = e^{2x}(2x+1)$, and that when I plug in $x=1/2$ then I get that the slope = $2e$.
So at this point I have the original curve's equation, the equation of its differential, the fact that the slope of the tangent at the given point is $2e$ and that this tangent also passes through the point $(\frac{1}{2}, \frac{e}{2})$. But I can't seem to arrive at the equation of this tangent.
$$y = 2ex - \frac{e}{2}$$
but how they got there, I don't know. I've checked other find the equation of a tangent line to a curve questions, but still haven't figured my way to that answer. It seems there's something wrong with my assumption that the equation of the tangent line is of the form $y=mx+c$. But how do I know which form it should take?
## Edit
Sorry - I'd written the target answer above wrong. I edited it to correct it.
-
Do you mean $2ex-e/2$? – Michael Albanese Jan 31 '13 at 10:58
I think it should be $2ex-\frac{e}{2}$ so that it passes through $(\frac{1}{2},\frac{e}{2})$. – Strin Jan 31 '13 at 11:13
That that you say is the answer can't possibly be correct as it is not the equation of a straight line... – DonAntonio Jan 31 '13 at 12:49
You're right... it was 2ex not e^x, I've edited the question. But even with this I seem to be making some fundamental mistake and can't arrive at it! – user60395 Jan 31 '13 at 13:43
The equation of a line of slope $m$ passing through a point $(x_0,y_0)$ is
$$y-y_0 = m (x - x_0)$$
Here, $m=2 e$, $x_0 = \frac{1}{2}$, and $y_0 = \frac{e}{2}$. Plug away.
-
so 1. F(x) = xe^2x
1. F(x)'= e^2x (2x + 1)
2. slope when you substitute x=1/2 = 2e
3. y - e/2 = 2e (x - 1/2)
y= 2ex - e + e/2
Y= 2ex - e/2
-
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# Composition of Linear Transformations is Linear Transformation
## Theorem
Let $K$ be a field.
Let $X, Y, Z$ be vector spaces over $K$.
Let $T_1 : X \to Y$ and $T_2 : Y \to Z$ be linear transformations.
Then the composition $T_2 \circ T_1 : X \to Z$ is a linear transformation.
## Proof
Let $\lambda \in K$ and $u, v \in X$.
Then, we have:
$\ds \map {\paren {T_2 \circ T_1} } {\lambda u + v}$ $=$ $\ds \map {T_2} {\map {T_1} {\lambda u + v} }$ $\ds$ $=$ $\ds \map {T_2} {\lambda T_1 u + T_1 v}$ Definition of Linear Transformation $\ds$ $=$ $\ds \lambda \paren {T_2 T_1} u + \paren {T_2 T_1} v$ $\ds$ $=$ $\ds \lambda \paren {T_2 \circ T_1} u + \paren {T_2 \circ T_1} v$
so $T_2 \circ T_1 : X \to Z$ is a linear transformation.
$\blacksquare$
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# physics question need help
Discussion in 'Homework Help' started by zelda1850, Jan 3, 2010.
1. ### zelda1850 Thread Starter New Member
Jan 3, 2010
18
0
A kicked soccer ball has an initial velocity of 25 meters per second at an angle of 40 degrees above the horizontal level ground [neglect friction]
A)calculate the magnitude of the vertical component of the balls initial velocity
b) calculate the maximum height of the ball reaches above its initial speed
c) sketch the balls height from the its initial position at point p until it returns to level ground
i understand how to do the first question but how can i figure out the maximum height for b?
Jul 3, 2008
2,433
469
3. ### zelda1850 Thread Starter New Member
Jan 3, 2010
18
0
emm heres my attempt
A) Viy sin 40 degrees
B) dy= vytrise is this the correct equation?
c) im not sure how i can sketch it yet
im confused with the 2nd problem how can i find maximum height if somoene can explain to me that wll help alot ^-^
4. ### thyristor Active Member
Dec 27, 2009
94
0
I think you know the answer to part a), just do the calculation.
A hint for part b): What will be the ball's vertical velocity when it reaches its maximum height?
A hint for part c): What well-known shape will the ball's trajectory display? (This is basic ballistics knowledge)
Jul 7, 2009
1,585
141
The aha! moment for many basic physics students with this problem comes from them realizing the proper model of the situation. Two key points are:
1. The horizontal motion is free particle motion, as there's no horizontal force on the ball after the kick.
2. The vertical motion is a pure kinematic one and you've no doubt already been given the constant acceleration (i.e., constant force) kinematic equations for this type of motion.
The third concept needed is the ability to resolve the initial velocity vector into the relevant components.
6. ### Fraser_Integration Member
Nov 28, 2009
142
5
Just write out all of the SUVAT (distance, init velocity, end velocity, acceleration, time) values you have for vertical and horizontal motion separately, and apply the any of the following formulas.
v = u + at
s = ut + 0.5at^2
v^2 = u^2 + 2as
7. ### hitmen Active Member
Sep 21, 2008
159
0
Uy = 25 sin 40 degrees
Vy = 0
g = -9.81ms-2
This could be good enough. Just sub in values
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# LeetCode Patching Array
Given a sorted positive integer array nums and an integer n, add/patch elements to the array such that any number in range `[1, n]`inclusive can be formed by the sum of some elements in the array. Return the minimum number of patches required.
Example 1:
nums = `[1, 3]`, n = `6`
Return `1`.
Combinations of nums are `[1], [3], [1,3]`, which form possible sums of: `1, 3, 4`.
Now if we add/patch `2` to nums, the combinations are: `[1], [2], [3], [1,3], [2,3], [1,2,3]`.
Possible sums are `1, 2, 3, 4, 5, 6`, which now covers the range `[1, 6]`.
So we only need `1` patch.
Example 2:
nums = `[1, 5, 10]`, n = `20`
Return `2`.
The two patches can be `[2, 4]`.
Example 3:
nums = `[1, 2, 2]`, n = `5`
Return `0`.
## Java Solution
The following algorithm is from https://discuss.leetcode.com/topic/35494/solution-explanation
We define `miss` to be the smallest number that will miss, which means, given an input {1…m}, we can build all sums in [0, miss). For the next number `x` in the array, if `x <= miss`, then we can build sums in the range `[0, miss + x)`, thus there is no need to add a number.
If the array is empty, or the next number `x` in the array is larger than `miss`, we have to add a number to get `miss`. The best strategy is to add `miss` itself. Then our next range is `[0, miss + miss)`
I also used an example below to describe how this method works.
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BREAKING NEWS
Force
## Summary
In physics, a force is an influence that can cause an object to change its velocity, i.e., to accelerate, meaning a change in speed or direction, unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a vector quantity. The SI unit of force is the newton (N), and force is often represented by the symbol F.[1]
Force
Forces can be described as a push or pull on an object. They can be due to phenomena such as gravity, magnetism, or anything that might cause a mass to accelerate.
Common symbols
${\displaystyle {\vec {F}}}$, F, F
SI unitnewton (N)
Other units
dyne, pound-force, poundal, kip, kilopond
In SI base unitskg·m·s−2
Derivations from
other quantities
F = ma
Dimension${\displaystyle {\mathsf {M}}{\mathsf {L}}{\mathsf {T}}^{-2}}$
Force plays a central role in classical mechanics, figuring in all three of Newton's laws of motion, which specify that the force on an object with an unchanging mass is equal to the product of the object's mass and the acceleration that it undergoes. Types of forces often encountered in classical mechanics include elastic, frictional, contact or "normal" forces, and gravitational. The rotational version of force is torque, which produces changes in the rotational speed of an object. In an extended body, each part often applies forces on the adjacent parts; the distribution of such forces through the body is the internal mechanical stress. In equilibrium these stresses cause no acceleration of the body as the forces balance one another. If these are not in equilibrium they can cause deformation of solid materials, or flow in fluids.
In modern physics, which includes relativity and quantum mechanics, the laws governing motion are revised to rely on fundamental interactions as the ultimate origin of force. However, the understanding of force provided by classical mechanics is useful for practical purposes.[2]
## Development of the concept
Philosophers in antiquity used the concept of force in the study of stationary and moving objects and simple machines, but thinkers such as Aristotle and Archimedes retained fundamental errors in understanding force. In part, this was due to an incomplete understanding of the sometimes non-obvious force of friction and a consequently inadequate view of the nature of natural motion.[3] A fundamental error was the belief that a force is required to maintain motion, even at a constant velocity. Most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Newton formulated laws of motion that were not improved for over two hundred years.[1]
By the early 20th century, Einstein developed a theory of relativity that correctly predicted the action of forces on objects with increasing momenta near the speed of light and also provided insight into the forces produced by gravitation and inertia. With modern insights into quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised a Standard Model to describe forces between particles smaller than atoms. The Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong, electromagnetic, weak, and gravitational.[4]: 2–10 [5]: 79 High-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction.[6]
## Pre-Newtonian concepts
Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was especially famous for formulating a treatment of buoyant forces inherent in fluids.[3]
Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the terrestrial sphere contained four elements that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed mostly of the elements earth and water, were in their natural place when on the ground, and that they stay that way if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g., for heavy bodies to fall), which led to "natural motion", and unnatural or forced motion, which required continued application of a force.[7] This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows. An archer causes the arrow to move at the start of the flight, and it then sails through the air even though no discernible efficient cause acts upon it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries the projectile to its target. This explanation requires a continuous medium such as air to sustain the motion.[8]
Though Aristotelian physics was criticized as early as the 6th century,[9][10] its shortcomings would not be corrected until the 17th century work of Galileo Galilei, who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion. He showed that the bodies were accelerated by gravity to an extent that was independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction.[11] Galileo's idea that force is needed to change motion rather than to sustain it, further improved upon by Isaac Beeckman, René Descartes, and Pierre Gassendi, became a key principle of Newtonian physics.[12]
In the early 17th century, before Newton's Principia, the term "force" (Latin: vis) was applied to many physical and non-physical phenomena, e.g., for an acceleration of a point. The product of a point mass and the square of its velocity was named vis viva (live force) by Leibniz. The modern concept of force corresponds to Newton's vis motrix (accelerating force).[13]
## Newtonian mechanics
Sir Isaac Newton described the motion of all objects using the concepts of inertia and force. In 1687, Newton published his magnum opus, Philosophiæ Naturalis Principia Mathematica.[1][14] In this work Newton set out three laws of motion that have dominated the way forces are described in physics to this day.[14] The precise ways in which Newton's laws are expressed have evolved in step with new mathematical approaches.[15]
### First law
Newton's first law of motion states that the natural behavior of an object at rest is to continue being at rest, and the natural behavior of an object moving at constant speed in a straight line is to continue moving at that constant speed along that straight line.[14] The latter follows from the former because of the principle that the laws of physics are the same for all inertial observers, i.e., all observers who do not feel themselves to be in motion. An observer moving in tandem with an object will see it as being at rest. So, its natural behavior will be to remain at rest with respect to that observer, which means that an observer who sees it moving at constant speed in a straight line will see it continuing to do so.[16]: 1–7
### Second law
According to the first law, motion at constant speed in a straight line does not need a cause. It is change in motion that requires a cause, and Newton's second law gives the quantitative relationship between force and change of motion. Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object.[17]: 204–207
A modern statement of Newton's second law is a vector equation:
${\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}},}$
where ${\displaystyle \mathbf {p} }$ is the momentum of the system, and ${\displaystyle \mathbf {F} }$ is the net (vector sum) force.[17]: 399 If a body is in equilibrium, there is zero net force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an unbalanced force acting on an object it will result in the object's momentum changing over time.[14]
In common engineering applications the mass in a system remains constant allowing as simple algebraic form for the second law. By the definition of momentum,
${\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}={\frac {\mathrm {d} \left(m\mathbf {v} \right)}{\mathrm {d} t}},}$
where m is the mass and ${\displaystyle \mathbf {v} }$ is the velocity.[4]: 9-1,9-2 If Newton's second law is applied to a system of constant mass, m may be moved outside the derivative operator. The equation then becomes
${\displaystyle \mathbf {F} =m{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}.}$
By substituting the definition of acceleration, the algebraic version of Newton's second law is derived:
${\displaystyle \mathbf {F} =m\mathbf {a} .}$
### Third law
Whenever one body exerts a force on another, the latter simultaneously exerts an equal and opposite force on the first. In vector form, if ${\displaystyle \mathbf {F} _{1,2}}$ is the force of body 1 on body 2 and ${\displaystyle \mathbf {F} _{2,1}}$ that of body 2 on body 1, then
${\displaystyle \mathbf {F} _{1,2}=-\mathbf {F} _{2,1}.}$
This law is sometimes referred to as the action-reaction law, with ${\displaystyle \mathbf {F} _{1,2}}$ called the action and ${\displaystyle -\mathbf {F} _{2,1}}$ the reaction.
Newton's Third Law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. The third law means that all forces are interactions between different bodies.[18][19] and thus that there is no such thing as a unidirectional force or a force that acts on only one body.
In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero:
${\displaystyle \mathbf {F} _{1,2}+\mathbf {F} _{2,1}=0.}$
More generally, in a closed system of particles, all internal forces are balanced. The particles may accelerate with respect to each other but the center of mass of the system will not accelerate. If an external force acts on the system, it will make the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system.[4]: 19-1 [5]
Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of a system is conserved in any closed system. In a system of two particles, if ${\displaystyle \mathbf {p} _{1}}$ is the momentum of object 1 and ${\displaystyle \mathbf {p} _{2}}$ the momentum of object 2, then
${\displaystyle {\frac {\mathrm {d} \mathbf {p} _{1}}{\mathrm {d} t}}+{\frac {\mathrm {d} \mathbf {p} _{2}}{\mathrm {d} t}}=\mathbf {F} _{1,2}+\mathbf {F} _{2,1}=0.}$
Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained.[4]: ch.12 [5]
### Defining "force"
Some textbooks use Newton's second law as a definition of force.[20][21][22][23] However, for the equation ${\displaystyle \mathbf {F} =m\mathbf {a} }$ for a constant mass ${\displaystyle m}$ to then have any predictive content, it must be combined with further information.[24][4]: 12-1 Moreover, inferring that a force is present because a body is accelerating is only valid in an inertial frame of reference.[5]: 59 The question of which aspects of Newton's laws to take as definitions and which to regard as holding physical content has been answered in various ways,[25][26]: vii which ultimately do not affect how the theory is used in practice.[25] Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force include Ernst Mach and Walter Noll.[27][28]
## Combining forces
Forces act in a particular direction and have sizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as "vector quantities". This means that forces follow a different set of mathematical rules than physical quantities that do not have direction (denoted scalar quantities). For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the result. If both of these pieces of information are not known for each force, the situation is ambiguous.[17]: 197
Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive vector quantities: they have magnitude and direction.[1] When two forces act on a point particle, the resulting force, the resultant (also called the net force), can be determined by following the parallelogram rule of vector addition: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector that is equal in magnitude and direction to the transversal of the parallelogram. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action.[4]: ch.12 [5]
Free-body diagrams can be used as a convenient way to keep track of forces acting on a system. Ideally, these diagrams are drawn with the angles and relative magnitudes of the force vectors preserved so that graphical vector addition can be done to determine the net force.[29]
As well as being added, forces can also be resolved into independent components at right angles to each other. A horizontal force pointing northeast can therefore be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Resolving force vectors into components of a set of basis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions.[30] This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. Orthogonal components are independent of each other because forces acting at ninety degrees to each other have no effect on the magnitude or direction of the other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one non-zero component. Orthogonal force vectors can be three-dimensional with the third component being at right angles to the other two.[4]: ch.12 [5]
### Equilibrium
When all the forces that act upon an object are balanced, then the object is said to be in a state of equilibrium.[17]: 566 Hence, equilibrium occurs when the resultant force acting on a point particle is zero (that is, the vector sum of all forces is zero). When dealing with an extended body, it is also necessary that the net torque be zero. A body is in static equilibrium with respect to a frame of reference if it at rest and not accelerating, whereas a body in dynamic equilibrium is moving at a constant speed in a straight line, i.e., moving but not accelerating. What one observer sees as static equilibrium, another can see as dynamic equilibrium and vice versa.[17]: 566
#### Static
Static equilibrium was understood well before the invention of classical mechanics. Objects that are not accelerating have zero net force acting on them.[31]
The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, a force is applied by the surface that resists the downward force with equal upward force (called a normal force). The situation produces zero net force and hence no acceleration.[1]
Pushing against an object that rests on a frictional surface can result in a situation where the object does not move because the applied force is opposed by static friction, generated between the object and the table surface. For a situation with no movement, the static friction force exactly balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object.[1]
A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as weighing scales and spring balances. For example, an object suspended on a vertical spring scale experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force", which equals the object's weight. Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded his Three Laws of Motion.[1][4]: ch.12 [5]
#### Dynamic
Dynamic equilibrium was first described by Galileo who noticed that certain assumptions of Aristotelian physics were contradicted by observations and logic. Galileo realized that simple velocity addition demands that the concept of an "absolute rest frame" did not exist. Galileo concluded that motion in a constant velocity was completely equivalent to rest. This was contrary to Aristotle's notion of a "natural state" of rest that objects with mass naturally approached. Simple experiments showed that Galileo's understanding of the equivalence of constant velocity and rest were correct. For example, if a mariner dropped a cannonball from the crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the cannonball fall straight down while the ship moved beneath it. Thus, in an Aristotelian universe, the falling cannonball would land behind the foot of the mast of a moving ship. When this experiment is actually conducted, the cannonball always falls at the foot of the mast, as if the cannonball knows to travel with the ship despite being separated from it. Since there is no forward horizontal force being applied on the cannonball as it falls, the only conclusion left is that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no force is required to keep the cannonball moving at the constant forward velocity.[11]
Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamic equilibrium: when all the forces on an object balance but it still moves at a constant velocity. A simple case of dynamic equilibrium occurs in constant velocity motion across a surface with kinetic friction. In such a situation, a force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in zero net force, but since the object started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle misinterpreted this motion as being caused by the applied force. When kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion.[4]: ch.12 [5]
## Examples of forces in classical mechanics
Some forces are consequences of the fundamental ones. In such situations, idealized models can be used to gain physical insight. For example, each solid object is considered a rigid body.[citation needed]
### Gravitational
What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the acceleration of every object in free-fall was constant and independent of the mass of the object. Today, this acceleration due to gravity towards the surface of the Earth is usually designated as ${\displaystyle \mathbf {g} }$ and has a magnitude of about 9.81 meters per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth.[32] This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of ${\displaystyle m}$ will experience a force:
${\displaystyle \mathbf {F} =m\mathbf {g} .}$
For an object in free-fall, this force is unopposed and the net force on the object is its weight. For objects not in free-fall, the force of gravity is opposed by the reaction forces applied by their supports. For example, a person standing on the ground experiences zero net force, since a normal force (a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward.[4]: ch.12 [5]
Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which Aristotle had assumed were in a natural state of constant motion, with falling motion observed on the Earth. He proposed a law of gravity that could account for the celestial motions that had been described earlier using Kepler's laws of planetary motion.[33]
Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body.[33] Combining these ideas gives a formula that relates the mass (${\displaystyle m_{\oplus }}$ ) and the radius (${\displaystyle R_{\oplus }}$ ) of the Earth to the gravitational acceleration:
${\displaystyle \mathbf {g} =-{\frac {Gm_{\oplus }}{{R_{\oplus }}^{2}}}{\hat {\mathbf {r} }},}$
where the vector direction is given by ${\displaystyle {\hat {\mathbf {r} }}}$ , is the unit vector directed outward from the center of the Earth.[14]
In this equation, a dimensional constant ${\displaystyle G}$ is used to describe the relative strength of gravity. This constant has come to be known as the Newtonian constant of gravitation, though its value was unknown in Newton's lifetime. Not until 1798 was Henry Cavendish able to make the first measurement of ${\displaystyle G}$ using a torsion balance; this was widely reported in the press as a measurement of the mass of the Earth since knowing ${\displaystyle G}$ could allow one to solve for the Earth's mass given the above equation. Newton realized that since all celestial bodies followed the same laws of motion, his law of gravity had to be universal. Succinctly stated, Newton's law of gravitation states that the force on a spherical object of mass ${\displaystyle m_{1}}$ due to the gravitational pull of mass ${\displaystyle m_{2}}$ is
${\displaystyle \mathbf {F} =-{\frac {Gm_{1}m_{2}}{r^{2}}}{\hat {\mathbf {r} }},}$
where ${\displaystyle r}$ is the distance between the two objects' centers of mass and ${\displaystyle {\hat {\mathbf {r} }}}$ is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.[14]
This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until the 20th century. During that time, sophisticated methods of perturbation analysis[34] were invented to calculate the deviations of orbits due to the influence of multiple bodies on a planet, moon, comet, or asteroid. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed.[35]
### Electromagnetic
The electrostatic force was first described in 1784 by Coulomb as a force that existed intrinsically between two charges.[36]: 519 The properties of the electrostatic force were that it varied as an inverse square law directed in the radial direction, was both attractive and repulsive (there was intrinsic polarity), was independent of the mass of the charged objects, and followed the superposition principle. Coulomb's law unifies all these observations into one succinct statement.[37]
Subsequent mathematicians and physicists found the construct of the electric field to be useful for determining the electrostatic force on an electric charge at any point in space. The electric field was based on using a hypothetical "test charge" anywhere in space and then using Coulomb's Law to determine the electrostatic force.[38]: 4-6–4-8 Thus the electric field anywhere in space is defined as
${\displaystyle \mathbf {E} ={\mathbf {F} \over {q}},}$
where ${\displaystyle q}$ is the magnitude of the hypothetical test charge. Similarly, the idea of the magnetic field was introduced to express how magnets can influence one another at a distance. The Lorentz force law gives the force upon a body with charge ${\displaystyle q}$ due to electric and magnetic fields:
${\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right),}$
where ${\displaystyle \mathbf {F} }$ is the electromagnetic force, ${\displaystyle \mathbf {E} }$ is the electric field at the body's location, ${\displaystyle \mathbf {B} }$ is the magnetic field, and ${\displaystyle \mathbf {v} }$ is the velocity of the particle. The magnetic contribution to the Lorentz force is the cross product of the velocity vector with the magnetic field.[39][40]: 482
The origin of electric and magnetic fields would not be fully explained until 1864 when James Clerk Maxwell unified a number of earlier theories into a set of 20 scalar equations, which were later reformulated into 4 vector equations by Oliver Heaviside and Josiah Willard Gibbs.[41] These "Maxwell's equations" fully described the sources of the fields as being stationary and moving charges, and the interactions of the fields themselves. This led Maxwell to discover that electric and magnetic fields could be "self-generating" through a wave that traveled at a speed that he calculated to be the speed of light. This insight united the nascent fields of electromagnetic theory with optics and led directly to a complete description of the electromagnetic spectrum.[42]
### Normal
When objects are in contact, the force directly between them is called the normal force, the component of the total force in the system exerted normal to the interface between the objects.[36]: 264 The normal force is closely related to Newton's third law. The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force on an object crashing into an immobile surface.[4]: ch.12 [5]
### Friction
Friction is a force that opposes relative motion of two bodies. At the macroscopic scale, the frictional force is directly related to the normal force at the point of contact. There are two broad classifications of frictional forces: static friction and kinetic friction.[17]: 267
The static friction force (${\displaystyle \mathbf {F} _{\mathrm {sf} }}$ ) will exactly oppose forces applied to an object parallel to a surface up to the limit specified by the coefficient of static friction (${\displaystyle \mu _{\mathrm {sf} }}$ ) multiplied by the normal force (${\displaystyle \mathbf {F} _{\text{N}}}$ ). In other words, the magnitude of the static friction force satisfies the inequality:
${\displaystyle 0\leq \mathbf {F} _{\mathrm {sf} }\leq \mu _{\mathrm {sf} }\mathbf {F} _{\mathrm {N} }.}$
The kinetic friction force (${\displaystyle F_{\mathrm {kf} }}$ ) is typically independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals:
${\displaystyle \mathbf {F} _{\mathrm {kf} }=\mu _{\mathrm {kf} }\mathbf {F} _{\mathrm {N} },}$
where ${\displaystyle \mu _{\mathrm {kf} }}$ is the coefficient of kinetic friction. The coefficient of kinetic friction is normally less than the coefficient of static friction.[17]: 267–271
### Tension
Tension forces can be modeled using ideal strings that are massless, frictionless, unbreakable, and do not stretch. They can be combined with ideal pulleys, which allow ideal strings to switch physical direction. Ideal strings transmit tension forces instantaneously in action–reaction pairs so that if two objects are connected by an ideal string, any force directed along the string by the first object is accompanied by a force directed along the string in the opposite direction by the second object.[43] By connecting the same string multiple times to the same object through the use of a configuration that uses movable pulleys, the tension force on a load can be multiplied. For every string that acts on a load, another factor of the tension force in the string acts on the load. Such machines allow a mechanical advantage for a corresponding increase in the length of displaced string needed to move the load. These tandem effects result ultimately in the conservation of mechanical energy since the work done on the load is the same no matter how complicated the machine.[4]: ch.12 [5][44]
### Spring
A simple elastic force acts to return a spring to its natural length. An ideal spring is taken to be massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push when contracted, or pull when extended, in proportion to the displacement of the spring from its equilibrium position.[45] This linear relationship was described by Robert Hooke in 1676, for whom Hooke's law is named. If ${\displaystyle \Delta x}$ is the displacement, the force exerted by an ideal spring equals:
${\displaystyle \mathbf {F} =-k\Delta \mathbf {x} ,}$
where ${\displaystyle k}$ is the spring constant (or force constant), which is particular to the spring. The minus sign accounts for the tendency of the force to act in opposition to the applied load.[4]: ch.12 [5]
### Centripetal
For an object in uniform circular motion, the net force acting on the object equals:[46]
${\displaystyle \mathbf {F} =-{\frac {mv^{2}}{r}}{\hat {\mathbf {r} }},}$
where ${\displaystyle m}$ is the mass of the object, ${\displaystyle v}$ is the velocity of the object and ${\displaystyle r}$ is the distance to the center of the circular path and ${\displaystyle {\hat {\mathbf {r} }}}$ is the unit vector pointing in the radial direction outwards from the center. This means that the net force felt by the object is always directed toward the center of the curving path. Such forces act perpendicular to the velocity vector associated with the motion of an object, and therefore do not change the speed of the object (magnitude of the velocity), but only the direction of the velocity vector. More generally, the net force that accelerates an object can be resolved into a component that is perpendicular to the path, and one that is tangential to the path. This yields both the tangential force, which accelerates the object by either slowing it down or speeding it up, and the radial (centripetal) force, which changes its direction.[4]: ch.12 [5]
### Continuum mechanics
Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealized point particles rather than three-dimensional objects. In real life, matter has extended structure and forces that act on one part of an object might affect other parts of an object. For situations where lattice holding together the atoms in an object is able to flow, contract, expand, or otherwise change shape, the theories of continuum mechanics describe the way forces affect the material. For example, in extended fluids, differences in pressure result in forces being directed along the pressure gradients as follows:
${\displaystyle {\frac {\mathbf {F} }{V}}=-\mathbf {\nabla } P,}$
where ${\displaystyle V}$ is the volume of the object in the fluid and ${\displaystyle P}$ is the scalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in the buoyant force for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight.[4]: ch.12 [5]
A specific instance of such a force that is associated with dynamic pressure is fluid resistance: a body force that resists the motion of an object through a fluid due to viscosity. For so-called "Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction:
${\displaystyle \mathbf {F} _{\mathrm {d} }=-b\mathbf {v} ,}$
where:
• ${\displaystyle b}$ is a constant that depends on the properties of the fluid and the dimensions of the object (usually the cross-sectional area), and
• ${\displaystyle \mathbf {v} }$ is the velocity of the object.[4]: ch.12 [5]
More formally, forces in continuum mechanics are fully described by a stress tensor with terms that are roughly defined as
${\displaystyle \sigma ={\frac {F}{A}},}$
where ${\displaystyle A}$ is the relevant cross-sectional area for the volume for which the stress tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the matrix diagonals of the tensor) as well as shear terms associated with forces that act parallel to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all strains (deformations) including also tensile stresses and compressions.[1][5]: 133–134 [38]: 38-1–38-11
### Fictitious
There are forces that are frame dependent, meaning that they appear due to the adoption of non-Newtonian (that is, non-inertial) reference frames. Such forces include the centrifugal force and the Coriolis force.[47] These forces are considered fictitious because they do not exist in frames of reference that are not accelerating.[4]: ch.12 [5] Because these forces are not genuine they are also referred to as "pseudo forces".[4]: 12-11
In general relativity, gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry.[48]
## Concepts derived from force
### Rotation and torque
Forces that cause extended objects to rotate are associated with torques. Mathematically, the torque of a force ${\displaystyle \mathbf {F} }$ is defined relative to an arbitrary reference point as the cross product:
${\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} ,}$
where ${\displaystyle \mathbf {r} }$ is the position vector of the force application point relative to the reference point.[17]: 497
Torque is the rotation equivalent of force in the same way that angle is the rotational equivalent for position, angular velocity for velocity, and angular momentum for momentum. As a consequence of Newton's first law of motion, there exists rotational inertia that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. Likewise, Newton's second law of motion can be used to derive an analogous equation for the instantaneous angular acceleration of the rigid body:
${\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }},}$
where
• ${\displaystyle I}$ is the moment of inertia of the body
• ${\displaystyle {\boldsymbol {\alpha }}}$ is the angular acceleration of the body.[17]: 502
This provides a definition for the moment of inertia, which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the tensor that, when properly analyzed, fully determines the characteristics of rotations including precession and nutation.[26]: 96–113
Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque:[49]
${\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {dt} }},}$
where ${\displaystyle \mathbf {L} }$ is the angular momentum of the particle.
Newton's Third Law of Motion requires that all objects exerting torques themselves experience equal and opposite torques,[50] and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques.
### Yank
The yank is defined as the rate of change of force[51]: 131
${\displaystyle \mathbf {Y} ={\frac {\mathrm {d} \mathbf {F} }{\mathrm {d} t}}}$
The term is used in biomechanical analysis,[52] athletic assessment[53] and robotic control.[54] The second (called "tug"), third ("snatch"), fourth ("shake"), and higher derivatives are rarely used.[51]
### Kinematic integrals
Forces can be used to define a number of physical concepts by integrating with respect to kinematic variables. For example, integrating with respect to time gives the definition of impulse:[55]
${\displaystyle \mathbf {J} =\int _{t_{1}}^{t_{2}}{\mathbf {F} \,\mathrm {d} t},}$
which by Newton's Second Law must be equivalent to the change in momentum (yielding the Impulse momentum theorem).
Similarly, integrating with respect to position gives a definition for the work done by a force:[4]: 13-3
${\displaystyle W=\int _{\mathbf {x} _{1}}^{\mathbf {x} _{2}}{\mathbf {F} \cdot {\mathrm {d} \mathbf {x} }},}$
which is equivalent to changes in kinetic energy (yielding the work energy theorem).[4]: 13-3
Power P is the rate of change dW/dt of the work W, as the trajectory is extended by a position change ${\displaystyle d\mathbf {x} }$ in a time interval dt:[4]: 13-2
${\displaystyle \mathrm {d} W={\frac {\mathrm {d} W}{\mathrm {d} \mathbf {x} }}\cdot \mathrm {d} \mathbf {x} =\mathbf {F} \cdot \mathrm {d} \mathbf {x} ,}$
so
${\displaystyle P={\frac {\mathrm {d} W}{\mathrm {d} t}}={\frac {\mathrm {d} W}{\mathrm {d} \mathbf {x} }}\cdot {\frac {\mathrm {d} \mathbf {x} }{\mathrm {d} t}}=\mathbf {F} \cdot \mathbf {v} ,}$
with ${\displaystyle \mathbf {v} =\mathrm {d} \mathbf {x} /\mathrm {d} t}$ the velocity.
### Potential energy
Instead of a force, often the mathematically related concept of a potential energy field is used. For instance, the gravitational force acting upon an object can be seen as the action of the gravitational field that is present at the object's location. Restating mathematically the definition of energy (via the definition of work), a potential scalar field ${\displaystyle U(\mathbf {r} )}$ is defined as that field whose gradient is equal and opposite to the force produced at every point:
${\displaystyle \mathbf {F} =-\mathbf {\nabla } U.}$
Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while nonconservative forces are not.[4]: ch.12 [5]
### Conservation
A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or potential forms. This means that for a closed system, the net mechanical energy is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space,[56] and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the contour map of the elevation of an area.[4]: ch.12 [5]
Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models that are dependent on a position often given as a radial vector ${\displaystyle \mathbf {r} }$ emanating from spherically symmetric potentials.[57] Examples of this follow:
For gravity:
${\displaystyle \mathbf {F} _{\text{g}}=-{\frac {Gm_{1}m_{2}}{r^{2}}}{\hat {\mathbf {r} }},}$
where ${\displaystyle G}$ is the gravitational constant, and ${\displaystyle m_{n}}$ is the mass of object n.
For electrostatic forces:
${\displaystyle \mathbf {F} _{\text{e}}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}r^{2}}}{\hat {\mathbf {r} }},}$
where ${\displaystyle \varepsilon _{0}}$ is electric permittivity of free space, and ${\displaystyle q_{n}}$ is the electric charge of object n.
For spring forces:
${\displaystyle \mathbf {F} _{\text{s}}=-kr{\hat {\mathbf {r} }},}$
where ${\displaystyle k}$ is the spring constant.[4]: ch.12 [5]
For certain physical scenarios, it is impossible to model forces as being due to a simple gradient of potentials. This is often due a macroscopic statistical average of microstates. For example, static friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, tension, compression, and drag. For any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.[4]: ch.12 [5]
The connection between macroscopic nonconservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the internal energies of the system, and are often associated with the transfer of heat. According to the Second law of thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases.[4]: ch.12 [5]
## Units
The SI unit of force is the newton (symbol N), which is the force required to accelerate a one kilogram mass at a rate of one meter per second squared, or kg·m·s−2.The corresponding CGS unit is the dyne, the force required to accelerate a one gram mass by one centimeter per second squared, or g·cm·s−2. A newton is thus equal to 100,000 dynes.[58]
The gravitational foot-pound-second English unit of force is the pound-force (lbf), defined as the force exerted by gravity on a pound-mass in the standard gravitational field of 9.80665 m·s−2.[58] The pound-force provides an alternative unit of mass: one slug is the mass that will accelerate by one foot per second squared when acted on by one pound-force.[58] An alternative unit of force in a different foot–pound–second system, the absolute fps system, is the poundal, defined as the force required to accelerate a one-pound mass at a rate of one foot per second squared.[58]
The pound-force has a metric counterpart, less commonly used than the newton: the kilogram-force (kgf) (sometimes kilopond), is the force exerted by standard gravity on one kilogram of mass. The kilogram-force leads to an alternate, but rarely used unit of mass: the metric slug (sometimes mug or hyl) is that mass that accelerates at 1 m·s−2 when subjected to a force of 1 kgf. The kilogram-force is not a part of the modern SI system, and is generally deprecated, sometimes used for expressing aircraft weight, jet thrust, bicycle spoke tension, torque wrench settings and engine output torque.[58]
Units of force
newton dyne kilogram-force,
kilopond
pound-force poundal
1 N ≡ 1 kg⋅m/s2 = 105 dyn ≈ 0.10197 kp ≈ 0.22481 lbf ≈ 7.2330 pdl
1 dyn = 10–5 N 1 g⋅cm/s2 1.0197×10−6 kp 2.2481×10−6 lbf 7.2330×10−5 pdl
1 kp = 9.80665 N = 980665 dyn gn × 1 kg 2.2046 lbf 70.932 pdl
1 lbf 4.448222 N 444822 dyn 0.45359 kp gn × 1 lb 32.174 pdl
1 pdl 0.138255 N 13825 dyn 0.014098 kp 0.031081 lbf 1 lb⋅ft/s2
The value of gn as used in the official definition of the kilogram-force (9.80665 m/s2) is used here for all gravitational units.
## Revisions of the force concept
At the beginning of the 20th century, new physical ideas emerged to explain experimental results in astronomical and submicroscopic realms. As discussed below, relativity alters the definition of momentum and quantum mechanics reuses the concept of "force" in microscopic contexts where Newton's laws do not apply directly.
### Special theory of relativity
In the special theory of relativity, mass and energy are equivalent (as can be seen by calculating the work required to accelerate an object). When an object's velocity increases, so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's Second Law,
${\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}},}$
remains valid because it is a mathematical definition.[36]: 855–876 But for momentum to be conserved at relativistic relative velocity, ${\displaystyle v}$ , momentum must be redefined as:
${\displaystyle \mathbf {p} ={\frac {m_{0}\mathbf {v} }{\sqrt {1-v^{2}/c^{2}}}},}$
where ${\displaystyle m_{0}}$ is the rest mass and ${\displaystyle c}$ the speed of light.
The expression relating force and acceleration for a particle with constant non-zero rest mass ${\displaystyle m}$ moving in the ${\displaystyle x}$ direction at velocity ${\displaystyle v}$ is:[59]: 216
${\displaystyle \mathbf {F} =\left(\gamma ^{3}ma_{x},\gamma ma_{y},\gamma ma_{z}\right),}$
where
${\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.}$
is called the Lorentz factor. The Lorentz factor increases steeply as the relative velocity approaches the speed of light. Consequently, the greater and greater force must be applied to produce the same acceleration at extreme velocity. The relative velocity cannot reach ${\displaystyle c}$ .[59]: 26 [4]: §15–8 If ${\displaystyle v}$ is very small compared to ${\displaystyle c}$ , then ${\displaystyle \gamma }$ is very close to 1 and
${\displaystyle \mathbf {F} =m\mathbf {a} }$
is a close approximation. Even for use in relativity, one can restore the form of
${\displaystyle F^{\mu }=mA^{\mu }}$
through the use of four-vectors. This relation is correct in relativity when ${\displaystyle F^{\mu }}$ is the four-force, ${\displaystyle m}$ is the invariant mass, and ${\displaystyle A^{\mu }}$ is the four-acceleration.[60]
The general theory of relativity incorporates a more radical departure from the Newtonian way of thinking about force, specifically gravitational force. This reimagining of the nature of gravity is described more fully below.
### Quantum mechanics
Quantum mechanics is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is different from that of classical physics. Instead of thinking about quantities like position, momentum, and energy as properties that an object has, one considers what result might appear when a measurement of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result.[61][62] The expectation value for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.[63]
In quantum mechanics, interactions are typically described in terms of energy rather than force. The Ehrenfest theorem provides a connection between quantum expectation values and the classical concept of force, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, the Born rule is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law, with a force defined as the negative derivative of the potential energy. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.[64][65]
Quantum mechanics also introduces two new constraints that interact with forces at the submicroscopic scale and which are especially important for atoms. Despite the strong attraction of the nucleus, the uncertainty principle limits the minimum extent of an electron probability distribution[66] and the Pauli exclusion principle prevents electrons from sharing the same probability distribution.[67] This gives rise to an emergent pressure known as degeneracy pressure. The dynamic equilibrium between the degeneracy pressure and the attractive electromagnetic force give atoms, molecules, liquids, and solids stability.[68]
### Quantum field theory
In modern particle physics, forces and the acceleration of particles are explained as a mathematical by-product of exchange of momentum-carrying gauge bosons. With the development of quantum field theory and general relativity, it was realized that force is a redundant concept arising from conservation of momentum (4-momentum in relativity and momentum of virtual particles in quantum electrodynamics). The conservation of momentum can be directly derived from the homogeneity or symmetry of space and so is usually considered more fundamental than the concept of a force. Thus the currently known fundamental forces are considered more accurately to be "fundamental interactions".[6]: 199–128
While sophisticated mathematical descriptions are needed to predict, in full detail, the result of such interactions, there is a conceptually simple way to describe them through the use of Feynman diagrams. In a Feynman diagram, each matter particle is represented as a straight line (see world line) traveling through time, which normally increases up or to the right in the diagram. Matter and anti-matter particles are identical except for their direction of propagation through the Feynman diagram. World lines of particles intersect at interaction vertices, and the Feynman diagram represents any force arising from an interaction as occurring at the vertex with an associated instantaneous change in the direction of the particle world lines. Gauge bosons are emitted away from the vertex as wavy lines and, in the case of virtual particle exchange, are absorbed at an adjacent vertex.[69] The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of fundamental interactions but are conceptually separate from forces can also be described using the same rules. For example, a Feynman diagram can describe in succinct detail how a neutron decays into an electron, proton, and antineutrino, an interaction mediated by the same gauge boson that is responsible for the weak nuclear force.[69]
## Fundamental interactions
All of the known forces of the universe are classified into four fundamental interactions. The strong and the weak forces act only at very short distances, and are responsible for the interactions between subatomic particles, including nucleons and compound nuclei. The electromagnetic force acts between electric charges, and the gravitational force acts between masses. All other forces in nature derive from these four fundamental interactions operating within quantum mechanics, including the constraints introduced by the Schrödinger equation and the Pauli exclusion principle.[67] For example, friction is a manifestation of the electromagnetic force acting between atoms of two surfaces. The forces in springs, modeled by Hooke's law, are also the result of electromagnetic forces. Centrifugal forces are acceleration forces that arise simply from the acceleration of rotating frames of reference.[4]: 12-11 [5]: 359
The fundamental theories for forces developed from the unification of different ideas. For example, Newton's universal theory of gravitation showed that the force responsible for objects falling near the surface of the Earth is also the force responsible for the falling of celestial bodies about the Earth (the Moon) and around the Sun (the planets). Michael Faraday and James Clerk Maxwell demonstrated that electric and magnetic forces were unified through a theory of electromagnetism. In the 20th century, the development of quantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter (fermions) interacting by exchanging virtual particles called gauge bosons.[70] This Standard Model of particle physics assumes a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in electroweak theory, which was subsequently confirmed by observation.[71]
The four fundamental forces of nature[72]
Property/Interaction Gravitation Weak Electromagnetic Strong
(Electroweak) Fundamental Residual
Acts on: Mass - Energy Flavor Electric charge Color charge Atomic nuclei
Particles experiencing: All Quarks, leptons Electrically charged Quarks, Gluons Hadrons
Particles mediating: Graviton
(not yet observed)
W+ W Z0 γ Gluons Mesons
Strength in the scale of quarks: 10−41 10−4 1 60 Not applicable
to quarks
Strength in the scale of
protons/neutrons:
10−36 10−7 1 Not applicable
20
### Gravitational
Newton's law of gravitation is an example of action at a distance: one body, like the Sun, exerts an influence upon any other body, like the Earth, no matter how far apart they are. Moreover, this action at a distance is instantaneous. According to Newton's theory, the one body shifting position changes the gravitational pulls felt by all other bodies, all at the same instant of time. Albert Einstein recognized that this was inconsistent with special relativity and its prediction that influences cannot travel faster than the speed of light. So, he sought a new theory of gravitation that would be relativistically consistent.[74][75] Mercury's orbit did not match that predicted by Newton's law of gravitation. Some astrophysicists predicted the existence of an undiscovered planet (Vulcan) that could explain the discrepancies. When Einstein formulated his theory of general relativity (GR) he focused on Mercury's problematic orbit and found that his theory added a correction, which could account for the discrepancy. This was the first time that Newton's theory of gravity had been shown to be inexact.[76]
Since then, general relativity has been acknowledged as the theory that best explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in gravitational fields travel under their own inertia in straight lines through curved spacetime – defined as the shortest spacetime path between two spacetime events. From the perspective of the object, all motion occurs as if there were no gravitation whatsoever. It is only when observing the motion in a global sense that the curvature of spacetime can be observed and the force is inferred from the object's curved path. Thus, the straight line path in spacetime is seen as a curved line in space, and it is called the ballistic trajectory of the object. For example, a basketball thrown from the ground moves in a parabola, as it is in a uniform gravitational field. Its spacetime trajectory is almost a straight line, slightly curved (with the radius of curvature of the order of few light-years). The time derivative of the changing momentum of the object is what we label as "gravitational force".[5]
### Electromagnetic
Maxwell's equations and the set of techniques built around them adequately describe a wide range of physics involving force in electricity and magnetism. This classical theory already includes relativity effects.[77] Understanding quantized electromagnetic interactions between elementary particles requires quantum electrodynamics (or QED). In QED, photons are fundamental exchange particles, describing all interactions relating to electromagnetism including the electromagnetic force.[78]
### Strong nuclear
There are two "nuclear forces", which today are usually described as interactions that take place in quantum theories of particle physics. The strong nuclear force is the force responsible for the structural integrity of atomic nuclei, and gains its name from its ability to overpower the electromagnetic repulsion between protons.[36]: 940 [79]
The strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD).[80] The strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. The strong force only acts directly upon elementary particles. A residual is observed between hadrons (notably, the nucleons in atomic nuclei), known as the nuclear force. Here the strong force acts indirectly, transmitted as gluons that form part of the virtual pi and rho mesons, the classical transmitters of the nuclear force. The failure of many searches for free quarks has shown that the elementary particles affected are not directly observable. This phenomenon is called color confinement.[81]: 232
### Weak nuclear
Unique among the fundamental interactions, the weak nuclear force creates no bound states.[82] The weak force is due to the exchange of the heavy W and Z bosons. Since the weak force is mediated by two types of bosons, it can be divided into two types of interaction or "vertices" — charged current, involving the electrically charged W+ and W bosons, and neutral current, involving electrically neutral Z0 bosons. The most familiar effect of weak interaction is beta decay (of neutrons in atomic nuclei) and the associated radioactivity.[36]: 951 This is a type of charged-current interaction. The word "weak" derives from the fact that the field strength is some 1013 times less than that of the strong force. Still, it is stronger than gravity over short distances. A consistent electroweak theory has also been developed, which shows that electromagnetic forces and the weak force are indistinguishable at a temperatures in excess of approximately 1015 K.[83] Such temperatures occurred in the plasma collisions in the early moments of the Big Bang.[82]: 201
## References
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The fragile markets were caught off-guard Friday by an unexp [#permalink]
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23 Jun 2013, 20:19
The fragile markets were caught off-guard Friday by an unexpected drop in retail sales for May, which renewed concerns about the pace of the recovery in the United States.
• which renewed concerns about the pace of the recovery in the United States.
• an event that renewed concerns about the pace of the recovery in the United States.
• and renewed concerns about the pace of the recovery in the United States.
• renewed concerns about the pace of the recovery in the United States.
• renewing concerns about the pace of the recovery in the United States.
Solution:
This problem tests your ability to distinguish between different types of modifiers. The underlined portion in the original stimulus is a relative clause because it begins with the relative pronoun “which.” A relative clause modifier must be immediately adjacent to the noun it modifies. Because it doesn’t make sense to modify “May,” answer (A) is incorrect.
Option (B) is an appositive modifier, but again, there is no noun it is renaming, so this construction is also incorrect.
Choice (C) applies the main subject “fragile markets” to the verb “renewed,” implying that the fragile markets themselves renewed concerns about the pace of the recovery. This changes the meaning of the sentence and therefore should be eliminated.
Answer (D) eliminates the subject attached to the verb “renewed” and therefore renders the sentence incoherent.
Option (E) correctly uses a participial modifier (“renewing concerns…”) at the end of the sentence to show that it modifies the action of the entire previous clause. Only a participial modifier (using a verb that typically ends in –ing or –ed) can function in this position.
I am strucked with the explanation of B,C,D and E.
Can some one shower more light, this question has so many Concepts.
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23 Jun 2013, 20:29
This question has been discussed the-fragile-markets-were-caught-off-guard-friday-by-an-146176.html?fl=similar .. If there any any more queries regarding the same please let us know...
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23 Jun 2013, 21:50
MacFauz wrote:
This question has been discussed the-fragile-markets-were-caught-off-guard-friday-by-an-146176.html.. If there any any more queries regarding the same please let us know...
MacFauz,
The link you have given doesnot work.
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23 Jun 2013, 21:57
Sorry for the lapse.. I've made the edit.. Please check if it works now..
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23 Jun 2013, 22:17
MacFauz wrote:
This question has been discussed the-fragile-markets-were-caught-off-guard-friday-by-an-146176.html?fl=similar .. If there any any more queries regarding the same please let us know...
I have at least 500 Such questions, I can Post 10-20 in a week.
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23 Jun 2013, 22:20
You defnitely can.... But chances are that most of the questions have already been discussed on the forum.. So you can do a quick serach for the question and if you don't find any you can post your questions to the forum...
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# Ugly Number I & II
Ugly Number I
Write a program to check whether a given number is an ugly number.
Ugly numbers are positive numbers whose prime factors only include `2, 3, 5`. For example, `6, 8` are ugly while `14` is not ugly since it includes another prime factor `7`.
Note that `1` is typically treated as an ugly number.
```public boolean isUgly(int num) {
if (num <= 0) {
return false;
}
if (num == 1) {
return true;
}
int[] factors = {2, 3, 5};
for (int i = 0; i < factors.length; i++) {
while (num % factors[i] == 0) {
num = num / factors[i];
}
}
return num == 1;
}```
Ugly Number II
Write a program to find the `n`-th ugly number.
Ugly numbers are positive numbers whose prime factors only include `2, 3, 5`. For example, `1, 2, 3, 4, 5, 6, 8, 9, 10, 12` is the sequence of the first `10` ugly numbers.
Note that `1` is typically treated as an ugly number.
https://leetcode.com/discuss/52716/o-n-java-solution
Solution:
O(n) time, one pass solution. O(n) space.
The ugly-number sequence is 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, … because every number can only be divided by 2, 3, 5, one way to look at the sequence is to split the sequence to three groups as below:
```(1) 1×2, 2×2, 3×2, 4×2, 5×2, …
(2) 1×3, 2×3, 3×3, 4×3, 5×3, …
(3) 1×5, 2×5, 3×5, 4×5, 5×5, …```
We can find that every subsequence is the ugly-sequence itself (1, 2, 3, 4, 5, …) multiply 2, 3, 5.
Then we use similar merge method as merge sort, to get every ugly number from the three subsequence.
Every step we choose the smallest one, and move one step after,including nums with same value.
``` public int nthUglyNumber(int n) {
int[] ugly = new int[n];
ugly[0] = 1;
int index2 = 0, index3 = 0, index5 = 0;
int factor2 = 2, factor3 = 3, factor5 = 5;
for (int i = 1; i < n; i++) {
int min = Math.min(Math.min(factor2, factor3), factor5);
ugly[i] = min;
if (factor2 == min) {
index2++;
factor2 = 2 * ugly[index2];
}
if (factor3 == min) {
index3++;
factor3 = 3 * ugly[index3];
}
if (factor5 == min) {
index5++;
factor5 = 5 * ugly[index5];
}
}
return ugly[n - 1];
}```
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# Hash Function
Hash Function
In data structure Hash, hash function is used to convert a string(or any other type) into an integer smaller than hash size and bigger or equal to zero. The objective of designing a hash function is to “hash” the key as unreasonable as possible. A good hash function can avoid collision as less as possible. A widely used hash function algorithm is using a magic number 33, consider any string as a 33 based big integer like follow:
hashcode(“abcd”) = (ascii(a) * 333 + ascii(b) * 332 + ascii(c) *33 + ascii(d)) % HASH_SIZE
= (97* 333 + 98 * 332 + 99 * 33 +100) % HASH_SIZE
= 3595978 % HASH_SIZE
here HASH_SIZE is the capacity of the hash table (you can assume a hash table is like an array with index 0 ~ HASH_SIZE-1).
Given a string as a key and the size of hash table, return the hash value of this key.f
Example
For key=”abcd” and size=100, return 78
Clarification
For this problem, you are not necessary to design your own hash algorithm or consider any collision issue, you just need to implement the algorithm as described.
Solution: Use a helper to avoid overflow. To calculate a*33%HASH_SIZE, instead of multiply by 33, we add a to itself 33 times. if(result+num>mod) we only add num-mod it to the result, otherwise, just need to add num to result. In this case, we are either adding or removing HASH_SIZE from the final result so it won’t overflow.
```public int hashCode(char[] key, int HASH_SIZE) {
int result = 0;
for (int i = 0; i < key.length; i++) {
result = helper(result, 33, HASH_SIZE);
result += key[i];
result %= HASH_SIZE;
}
return result;
}
int helper(int num, int base, int mod) {
int result = 0;
for (int i = 0; i < base; i++) {
if (result + num > mod) {
result += num - mod;
} else {
result += num;
}
}
return result;
}```
# Leetcode: Valid Number
Validate if a given string is numeric.
Some examples:
`"0"` => `true`
`" 0.1 "` => `true`
`"abc"` => `false`
`"1 a"` => `false`
`"2e10"` => `true`
Note: It is intended for the problem statement to be ambiguous. You should gather all requirements up front before implementing one.
```public boolean isNumber(String s) {
// Start typing your Java solution below
// DO NOT write main() function
if(s==null||s.length()==0){
return false;
}
//continuous zeros from beginning since begin is true
int cont = 0;
boolean hasDot = false;
boolean begin = false;
boolean end = false;
boolean hasSign = false;
boolean hasE = false;
for(int i=0;i<s.length();i++){
char c = s.charAt(i);
if(c=='+'||c=='-'){
if(hasSign||begin){
return false;
}else{
hasSign = true;
}
continue;
}
if(c=='e'){
if(hasE||!begin){
return false;
}else{
return isInteger(s.substring(i+1));
//return true;
}
}
if(c=='.'){
if(hasDot||end){
return false;
}else{
hasDot = true;
}
continue;
}
if(c==' '){
if((begin ||hasSign) && !end){
end=true;
}
continue;
}
if(!(isDigit(s,i))){
return false;
}else{
if(end){
return false;
}else{
if(!begin){
begin=true;
}
}
}
}
if(!begin){
return false;
}
return true;
}
public boolean isInteger(String s){
if(s==null||s.length()==0){
return false;
}
if(s.length()==1 && isDigit(s,0)){
return true;
}
char c = s.charAt(0);
if(c=='0'){
return ifAllSpace(s.substring(1));
}
int zeors = 0;
for(int i=0;i<s.length();i++){
if(i==0){
if(c=='+'||c=='-'){
return isIntegerWithoutSign(s.substring(i+1));
}else
if(!isDigit(s,i)){
return false;
}
}else
if(!isDigit(s,i)){
if(c==' '){
return ifAllSpace(s.substring(i+1));
}else{
return false;
}
}
}
return true;
}
public boolean ifAllSpace(String s){
for(int i=0;i<s.length();i++){
if(s.charAt(i)!=' '){
return false;
}
}
return true;
}
public boolean isIntegerWithoutSign(String s){
if(s==null||s.length()==0){
return false;
}
if(s.length()==1 && isDigit(s,0)){
return true;
}
char c = s.charAt(0);
if(c=='0'){
return false;
}
for(int i=0;i<s.length();i++){
if(!isDigit(s,i)){
if(c==' '){
return ifAllSpace(s.substring(i));
}else{
return false;
}
}
}
return true;
}
public boolean isDigit(String s,int index){
char c = s.charAt(index);
if(c<(int)'0' || c>(int)'9'){
return false;
}
return true;
}```
# Leetcode: Permutation Sequence
The set `[1,2,3,…,n]` contains a total of n! unique permutations.
By listing and labeling all of the permutations in order,
We get the following sequence (ie, for n = 3):
1. `"123"`
2. `"132"`
3. `"213"`
4. `"231"`
5. `"312"`
6. `"321"`
Given n and k, return the kth permutation sequence.
Note: Given n will be between 1 and 9 inclusive.
```public String getPermutation(int n, int k) {
// Start typing your Java solution below
// DO NOT write main() function
ArrayList<Integer> nums = new ArrayList<Integer>();
for(int i=1;i<=n;i++){
}
k--;//k originally starts from 1
int index=0;
while(k!=0){
int frac = frac(n-index-1);
int targetNumIndex = k/frac+index;
int targetNum = nums.remove(targetNumIndex);
index++;
k=k%frac;
}
String output = "";
for(Integer num:nums){
output+=num;
}
return output;
}
public int frac(int n){
int sum = 1;
for(int i=1;i<=n;i++){
sum*=i;
}
return sum;
}```
# Leetcode: Roman to Integer
Given a roman numeral, convert it to an integer.
Input is guaranteed to be within the range from 1 to 3999.
```public int romanToInt(String s) {
// Start typing your Java solution below
// DO NOT write main() function
ArrayList<Character> oneLetters = new ArrayList<Character>();
ArrayList<Character> fiveLetters = new ArrayList<Character>();
int num=0;
for(int i=0;i<s.length();i++){
int bit = fiveLetters.indexOf(s.charAt(i));
//if the current letter is in fiveLetters
if(bit!=-1){
num+=5*(int)Math.pow(10,bit);
}else{
bit = oneLetters.indexOf(s.charAt(i));
//if the current letter is in oneLetters && there is a bigger number afterwards
if( i+1<s.length() &&
(fiveLetters.indexOf(s.charAt(i+1))>=bit ||
oneLetters.indexOf(s.charAt(i+1))>bit)){
num-=(int)Math.pow(10,bit);
}
else{
num+=(int)Math.pow(10,bit);
}
}
}
return num;
}```
# Leetcode: Integer to Roman
Given an integer, convert it to a roman numeral.
Input is guaranteed to be within the range from 1 to 3999.
```public String intToRoman(int num) {
// Start typing your Java solution below
// DO NOT write main() function
String[] oneLetters = {"I","X","C","M"};
String[] fiveLetters = {"V","L","D",""};
String numStr = num+"";
String output = "";
for(int i=numStr.length()-1;i>=0;i--){
String tmp = "";
int bit = (int)num%(int)Math.pow(10,i+1)/(int)Math.pow(10,i);
switch (bit){
case 1: tmp = oneLetters[i];
break;
case 2: tmp = oneLetters[i]+oneLetters[i];
break;
case 3: tmp = oneLetters[i]+oneLetters[i]+oneLetters[i];
break;
case 4: tmp = oneLetters[i]+fiveLetters[i];
break;
case 5: tmp = fiveLetters[i];
break;
case 6: tmp = fiveLetters[i]+oneLetters[i];
break;
case 7: tmp = fiveLetters[i]+oneLetters[i]+oneLetters[i];
break;
case 8: tmp = fiveLetters[i]+oneLetters[i]+oneLetters[i]+oneLetters[i];
break;
case 9: tmp = oneLetters[i]+oneLetters[i+1];
break;
default: tmp = "";
break;
}
output+=tmp;
}
return output;
}```
# Leetcode: Pow(x, n)
Implement pow(xn).
```public double pow(double x, int n) {
// Start typing your Java solution below
// DO NOT write main() function
boolean positive = n >= 0 ? true : false;
int nabs = n == Integer.MIN_VALUE ? Integer.MAX_VALUE : Math.abs(n);
double product = n == Integer.MIN_VALUE ? powHelp(x, nabs) * x : powHelp(x, nabs);
return positive ? product : 1 / product;
}
public double powHelp(double base, int n) {
if (n == 0) {
return 1;
}
if (n == 1) {
return base;
}
double tmp = powHelp(base, n / 2);
return tmp * tmp * powHelp(base, n % 2);
}```
O(logN)
Version with no helper function
```public double myPow(double x, int n) {
if(n==0){
return 1;
}
int n_abs = Math.abs(n);
if(n_abs==1){
return n>0?x:1/x;
}
double tmp = myPow(x, n/2);
return tmp*tmp*myPow(x, n%2);
}
```
# Leetcode: String to Integer (atoi)
Implement atoi to convert a string to an integer.
Hint: Carefully consider all possible input cases. If you want a challenge, please do not see below and ask yourself what are the possible input cases.
Notes: It is intended for this problem to be specified vaguely (ie, no given input specs). You are responsible to gather all the input requirements up front.
Requirements for atoi:The function first discards as many whitespace characters as necessary until the first non-whitespace character is found. Then, starting from this character, takes an optional initial plus or minus sign followed by as many numerical digits as possible, and interprets them as a numerical value.
The string can contain additional characters after those that form the integral number, which are ignored and have no effect on the behavior of this function.
If the first sequence of non-whitespace characters in str is not a valid integral number, or if no such sequence exists because either str is empty or it contains only whitespace characters, no conversion is performed.
If no valid conversion could be performed, a zero value is returned. If the correct value is out of the range of representable values, INT_MAX (2147483647) or INT_MIN (-2147483648) is returned.
```public class Solution {
public int atoi(String str) {
// Start typing your Java solution below
// DO NOT write main() function
ArrayList<Integer> validBits = new ArrayList<Integer>();
int sign=0;
if(str==null||str.length()==0){
return 0;
}
int index = 0;
while(str.charAt(index)==' '){
index++;
}
if(!(isNum(str, index) || str.charAt(index)=='+' || str.charAt(index)=='-')){
return 0;
}
if(str.charAt(index)=='-'){
sign = -1;
index++;
}else{
sign = 1;
if(str.charAt(index)=='+'){
index++;
}
}
//now find the first non-zero bit
while(str.charAt(index)=='0'){
index++;
}
while(index<str.length() && isNum(str,index)){
int asc = (int)str.charAt(index);
int bit = asc-'0';
index++;
}
int length = validBits.size();
if(length==0){
return 0;
}
double sum =0;
for(int i=0;i<length;i++){
sum+=validBits.get(i)*Math.pow(10,length-i-1);
}
if(sign<0){
if(sum*sign<(double)Integer.MIN_VALUE){
return Integer.MIN_VALUE;
}else{
return (int)(sum*sign);
}
}else{
if(sum>(double)Integer.MAX_VALUE){
return Integer.MAX_VALUE;
}else{
return (int)sum;
}
}
}
public boolean isNum(String str, int index){
int asc = (int)str.charAt(index);
if(asc>='0' && asc<='9'){
return true;
}
return false;
}
}```
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# Convert 9 metric tons (or tonnes) to milligrams (9 t to mg conversion)
## How much is 9 metric tons (or tonnes) in milligrams?
The answer is 9 metric tons (or tonnes) = 9000000000 milligrams. In other words, 9 metric tons (or tonnes) is equal to 9000000000 milligrams.
## 9 metric tons (or tonnes) to milligrams converter
Easily convert metric tons (or tonnes) to milligrams using the converter below. Simply enter the weight in metric tons (or tonnes) and the converter will display its equivalent in milligrams.
=
## How to convert 9 metric tons (or tonnes) to milligrams?
Since 1 metric ton (or tonne) = 1000000000 milligrams, to convert 9 metric tons (or tonnes) to milligrams, multiply 9 by 1000000000
### 9 metric tons (or tonnes) to milligrams conversion formula
The formula to convert 9 metric tons (or tonnes) to milligrams is given below:
milligrams = metric tons (or tonnes) × 1000000000
Below is a step-by-step explanation demonstrating how to use the formula for converting 9 t to mg:
milligrams = metric tons (or tonnes) × 1000000000
milligrams = 9 × 1000000000
milligrams = 9000000000
What is 9 t in mg? 9 t = 9000000000 mg
### 9 metric tons (or tonnes) to milligrams conversion factor
The conversion factor to convert 9 metric tons (or tonnes) to milligrams is 1000000000
## 9 metric tons (or tonnes) to milligrams conversion table
How many milligrams in 9 metric tons (or tonnes)? There are 9000000000 milligrams in 9 metric tons (or tonnes). The metric tons (or tonnes) to milligrams conversion chart below shows a list of various metric ton (or tonne) values converted to milligrams.
Metric tons (or tonnes) (t) Milligrams (mg)
9 t 9000000000 mg
9.01 t 9010000000 mg
9.02 t 9020000000 mg
9.03 t 9030000000 mg
9.04 t 9040000000 mg
9.05 t 9050000000 mg
9.06 t 9060000000 mg
9.07 t 9070000000 mg
9.08 t 9080000000 mg
9.09 t 9090000000 mg
9.1 t 9100000000 mg
9.11 t 9110000000 mg
9.12 t 9120000000 mg
9.13 t 9130000000 mg
9.14 t 9140000000 mg
9.15 t 9150000000 mg
9.16 t 9160000000 mg
9.17 t 9170000000 mg
9.18 t 9180000000 mg
9.19 t 9190000000 mg
9.2 t 9200000000 mg
9.21 t 9210000000 mg
9.22 t 9220000000 mg
9.23 t 9230000000 mg
9.24 t 9240000000 mg
9.25 t 9250000000 mg
9.26 t 9260000000 mg
9.27 t 9270000000 mg
9.28 t 9280000000 mg
9.29 t 9290000000 mg
9.3 t 9300000000 mg
9.31 t 9310000000 mg
9.32 t 9320000000 mg
9.33 t 9330000000 mg
9.34 t 9340000000 mg
9.35 t 9350000000 mg
9.36 t 9360000000 mg
9.37 t 9370000000 mg
9.38 t 9380000000 mg
9.39 t 9390000000 mg
9.4 t 9400000000 mg
9.41 t 9410000000 mg
9.42 t 9420000000 mg
9.43 t 9430000000 mg
9.44 t 9440000000 mg
9.45 t 9450000000 mg
9.46 t 9460000000 mg
9.47 t 9470000000 mg
9.48 t 9480000000 mg
9.49 t 9490000000 mg
9.5 t 9500000000 mg
9.51 t 9510000000 mg
9.52 t 9520000000 mg
9.53 t 9530000000 mg
9.54 t 9540000000 mg
9.55 t 9550000000 mg
9.56 t 9560000000 mg
9.57 t 9570000000 mg
9.58 t 9580000000 mg
9.59 t 9590000000 mg
9.6 t 9600000000 mg
9.61 t 9610000000 mg
9.62 t 9620000000 mg
9.63 t 9630000000 mg
9.64 t 9640000000 mg
9.65 t 9650000000 mg
9.66 t 9660000000 mg
9.67 t 9670000000 mg
9.68 t 9680000000 mg
9.69 t 9690000000 mg
9.7 t 9700000000 mg
9.71 t 9710000000 mg
9.72 t 9720000000 mg
9.73 t 9730000000 mg
9.74 t 9740000000 mg
9.75 t 9750000000 mg
9.76 t 9760000000 mg
9.77 t 9770000000 mg
9.78 t 9780000000 mg
9.79 t 9790000000 mg
9.8 t 9800000000 mg
9.81 t 9810000000 mg
9.82 t 9820000000 mg
9.83 t 9830000000 mg
9.84 t 9840000000 mg
9.85 t 9850000000 mg
9.86 t 9860000000 mg
9.87 t 9870000000 mg
9.88 t 9880000000 mg
9.89 t 9890000000 mg
9.9 t 9900000000 mg
9.91 t 9910000000 mg
9.92 t 9920000000 mg
9.93 t 9930000000 mg
9.94 t 9940000000 mg
9.95 t 9950000000 mg
9.96 t 9960000000 mg
9.97 t 9970000000 mg
9.98 t 9980000000 mg
9.99 t 9990000000 mg
## What is a metric ton?
A metric ton or tonne, is a unit of mass in the metric system. A metric ton is equal to 1000 kilograms or approximately 2204.62 pounds. The metric ton is commonly used worldwide as a standard unit for expressing mass in various contexts, such as international trade, manufacturing, and transportation. The metric ton is abbreviated using the symbol "t".
## What is a milligram?
A milligram (or milligramme) is a unit of mass in the International System of Units (SI). The SI prefix "milli-" denotes a factor of one thousandth, making a milligram equivalent to one thousandth of a gram. The milligram is used to measure the mass of small quantities of substances, including pharmaceuticals, chemicals, biological samples, and dosages of medications and supplements, as well as ingredients for recipes. The milligram is abbreviated using the symbol "mg".
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# Difference between revisions of "2018 AMC 12A Problems/Problem 19"
## Problem
Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum $$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots$$of the reciprocals of the elements of $A$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A)} \text{ 16} \qquad \textbf{(B)} \text{ 17} \qquad \textbf{(C)} \text{ 19} \qquad \textbf{(D)} \text{ 23} \qquad \textbf{(E)} \text{ 36}$
## Solution
It's just $$\sum_{a=0}^\infty\frac1{2^a}\sum_{b=0}^\infty\frac1{3^b}\sum_{c=0}^\infty\frac{1}{5^c} =\sum_{a=0}^\infty\sum_{b=0}^\infty\sum_{c=0}^\infty\frac1{2^a3^b5^c} = 2 \cdot \frac32 \cdot \frac54 = \frac{15}{4}\Rightarrow\textbf{(C)}.$$ since this represents all the numbers in the denominator. (athens2016)
## Solution 2
Separate into 7 separate infinite series's so we can calculate each and find the original sum:
The first infinite sequence shall be all the reciprocals of the powers of $2$, the second shall be reciprocals of the powers of $3$, and the third will consist of reciprocals of the powers of 5. We can easily calculate these to be $1, \frac{1}{2}, \frac{1}{4}$ respectively.
The fourth infinite series shall be all real numbers in the form $\frac{1}{2^a3^b}$, where $a$ and $b$ are greater than or equal to 1.
The fifth is all real numbers in the form $\frac{1}{2^a5^b}$, where $a$ and $b$ are greater than or equal to 1.
The sixth is all real numbers in the form $\frac{1}{3^a5^b}$, where $a$ and $b$ are greater than or equal to 1.
The seventh infinite series is all real numbers in the form $\frac{1}{2^a3^b5^c}$, where $a$ and $b$ and $c$ are greater than or equal to 1.
Let us denote the first sequence as $a_{1}$, the second as $a_{2}$, etc. We know $a_{1}=1$, $a_{2}=\frac{1}{2}$, $a_{3}=\frac{1}{4}$, let us find $a_{4}$. factoring out $\frac{1}{6}$ from the terms in this subsequence, we would get $a_{4}=\frac{1}{6}(1+a_{1}+a_{2}+a_{4})$.
Knowing $a_{1}$ and $a_{2}$, we can substitute and solve for $a_{4}$, and we get $\frac{1}{2}$. If we do similar procedures for the fifth and sixth sequences, we can solve for them too, and we get after solving them $\frac{1}{4}$ and $\frac{1}{8}$.
Finally, for the seventh sequence, we see $a_{7}=\frac{1}{30}(a_{8})$, where $a_{8}$ is the infinite series the problem is asking us to solve for. The sum of all seven subsequences will equal the one we are looking for, so solving, we get $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{a_{8}}{30}(a_{8})=a_{8}$, but when we separated the sequence into its parts, we ignored the $1/1$, so adding in the $1$, we get $1+1+\frac{1}{2}+\frac{1}{4}+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{a_{8}}{30}=a_{8}$, which when we solve for, we get $\frac{29}{8}=\frac{29a_{8}}{30}$, $\frac{1}{8}=\frac{a_{8}}{30}$, $\frac{30}{8}=(a_{8})$, $\frac{15}{4}=(a_{8})$. So our answer is $\frac{15}{4}$, but we are asked to add the numerator and denominator, which sums up to $19$, which is the answer.
~~Edited by mprincess0229~~
## See Also
2018 AMC 12A (Problems • Answer Key • Resources) Preceded byProblem 18 Followed byProblem 20 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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# Review: Independent variable- the x coordinate, the input (time will always be the independent) Dependent variable- the y coordinate, the output (need.
## Presentation on theme: "Review: Independent variable- the x coordinate, the input (time will always be the independent) Dependent variable- the y coordinate, the output (need."— Presentation transcript:
Review: Independent variable- the x coordinate, the input (time will always be the independent) Dependent variable- the y coordinate, the output (need the x value to determine) 1.) A woman who wears a size 8 shoe has a foot that is 10 inches long. A woman who wears a size 4 shoe has a foot that is 5 inches long. Name the independent variable and the dependent variable. Then write two ordered pairs. 2.) A 5 minute overseas call to Paris costs \$4.50. A ten minute overseas call to Paris costs \$8.50. Predicting Using Graphs
Elaine is saving \$12 a week to buy a new stereo. After 6 weeks, she has \$97 saved. (w,m) a). independent variable: b). dependent variable: c). slope (rate of change): d). y-intercept: e). Write the equation of the line: f). How much money will she have saved after 10 weeks? g.) What does the y-intercept represent? What does the slope represent?
A linear equation can be used to describe the length of a spring, y, when it is stretched by weight, x. A spring is 2 cm long when a 14-g weight is attached and 5 cm long when a 20-g weight is attached. a). independent variable: b). dependent variable: c). slope:(write two points to help) d). y-intercept: e). equation of the line: f). How long is the spring when a 16-g weight is attached?
A woman who wears a size 8 shoe has a foot that is 10 inches long. A woman who wears a size 4 shoe has a foot that is 5 inches long. a.) slope:(write two points to help) b). y-intercept: c). equation of the line: d.) How long is a foot that is a size 7 shoe?
A 5 minute overseas call to Paris costs \$4.50. A ten minute overseas call to Paris costs \$8.50. a.) slope:(write two points to help) b). y-intercept: c). equation of the line: d.) How much does a 2 minute call to Paris cost? e.) What does the y-intercept represent? What does the slope represent?
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If x and y are integers and xy 0, is x - y > 0 ? : GMAT Data Sufficiency (DS)
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# If x and y are integers and xy 0, is x - y > 0 ?
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27 Oct 2010, 17:38
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If x and y are integers and xy ≠ 0, is x - y > 0 ?
(1) x/y < 1/2
(2) $$\sqrt{x^2}=x$$ and $$\sqrt{y^2}=y$$
[Reveal] Spoiler:
Why I alone isn't sufficient , since x/y <1/2 => x<y/2 .. As X is even less than half of Y so X can't be greater than Y
[Reveal] Spoiler: OA
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Last edited by Bunuel on 11 Apr 2014, 02:45, edited 1 time in total.
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Re: integers x & y [#permalink]
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27 Oct 2010, 17:52
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shrive555 wrote:
If x and y are integers and xy ≠ 0, is x - y > 0 ?
1 - x/y <1/2
2- sqr X^2 = X and sqr Y^2 = Y
Why I alone isn't sufficient , since x/y <1/2 => x<y/2 .. As X is even less than half of Y so X can't be greater than Y
Question: is $$x>y$$?
(1) $$\frac{x}{y}<\frac{1}{2}$$ --> if both $$x$$ and $$y$$ are positive (for example 1 and 3 respectively) then $$x<y$$ and the answer to the question is NO but if $$x$$ and $$y$$ are both negative (for example -1 and -3 respectively) then $$x>y$$ and the answer to the question is YES. Not sufficient.
(2) $$\sqrt{x^2}=x$$ and $$\sqrt{y^2}=y$$ --> both $$x$$ and $$y$$ are positive (as square root function can not give negative result and we know that neither of unknown is zero), but we don't know whether $$x>y$$. Not sufficient.
(1)+(2) From (2) both $$x$$ and $$y$$ are positive so from (1) $$x<y$$ and the answer to the question is NO. Sufficient.
As for your solution: red part is not correct. You can not multiply inequality by y as you don't know the sign of it.
Never multiply or reduce inequality by an unknown (a variable) unless you are sure of its sign since you do not know whether you must flip the sign of the inequality.
Hope it helps.
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Re: integers x & y [#permalink]
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27 Oct 2010, 18:37
....Great Point to remember !!! +2
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Re: integers x & y [#permalink]
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27 Oct 2010, 23:52
Really good one Bunuel... thanks
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Re: If x and y are integers and xy 0, is x - y > 0 ? [#permalink]
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07 Feb 2014, 07:17
x and y are non-zero integers. Is x > y?
1. x/y < 1/2 -> x = 1 , y = 8, NO. x = -1, y = -8, YES.
2. |x| = x and |y| = y; x and y are both positive. but x > y and x < y possible. Remember Sqrt(x^2) = |x| and NOT x.
Combining, x and y are positive and x < y/2 => x < y.
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Re: If x and y are integers and xy 0, is x - y > 0 ? [#permalink]
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10 Apr 2014, 02:43
shrive555 wrote:
If x and y are integers and xy ≠ 0, is x - y > 0 ?
(1) x/y < 1/2
(2) $$\sqrt{x^2}=x$$
Why I alone isn't sufficient , since x/y <1/2 => x<y/2 .. As X is even less than half of Y so X can't be greater than Y
Please note some how statement 2 has been edited , part of statement 2 is missing, $$sqrt (y^2) = y$$, should be there.
The question as given now , gives answer as E.
Hope concerned persons will look into this.
Thank you.
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Re: If x and y are integers and xy 0, is x - y > 0 ? [#permalink]
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10 Apr 2014, 04:57
stne wrote:
shrive555 wrote:
If x and y are integers and xy ≠ 0, is x - y > 0 ?
(1) x/y < 1/2
(2) $$\sqrt{x^2}=x$$
Why I alone isn't sufficient , since x/y <1/2 => x<y/2 .. As X is even less than half of Y so X can't be greater than Y
Please note some how statement 2 has been edited , part of statement 2 is missing, $$sqrt (y^2) = y$$, should be there.
The question as given now , gives answer as E.
Hope concerned persons will look into this.
Thank you.
Hi Stne,
We do not really need $$sqrt (y^2) = y$$ because if $$\sqrt{x^2}=x$$ is true, x is positive
and as x/y < 1/2 => 2x < y
Hence as y is greater than 2x, it has to be positive.
*press Kudos if you like the post!
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Re: If x and y are integers and xy 0, is x - y > 0 ? [#permalink]
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10 Apr 2014, 07:05
ricsingh wrote:
stne wrote:
shrive555 wrote:
If x and y are integers and xy ≠ 0, is x - y > 0 ?
(1) x/y < 1/2
(2) $$\sqrt{x^2}=x$$
Why I alone isn't sufficient , since x/y <1/2 => x<y/2 .. As X is even less than half of Y so X can't be greater than Y
Please note some how statement 2 has been edited , part of statement 2 is missing, $$sqrt (y^2) = y$$, should be there.
The question as given now , gives answer as E.
Hope concerned persons will look into this.
Thank you.
Hi Stne,
We do not really need $$sqrt (y^2) = y$$ because if $$\sqrt{x^2}=x$$ is true, x is positive
and as x/y < 1/2 => 2x < y
Hence as y is greater than 2x, it has to be positive.
*press Kudos if you like the post!
You cannot cross multiply in inequality without knowing the signs of x and y , you can cross multiply only when you know that the signs are positive
if the question does not have $$sqrt (y^2) = y$$ then y need not necessarily be positive
consider x=1 and y = 3 this satisfies both the statement and the answer is no
consider x= 1 and y = -3 this satisfies both the statements and the answer is yes
as you can see, Y has to be positive for the answer to be c .
Hence question should be corrected, as in the present format, answer is E
Hope it helps , let me know if there is anything still unclear.
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Re: If x and y are integers and xy 0, is x - y > 0 ? [#permalink]
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10 Apr 2014, 07:44
stne wrote:
ricsingh wrote:
shrive555 wrote:
If x and y are integers and xy ≠ 0, is x - y > 0 ?
(1) x/y < 1/2
(2) $$\sqrt{x^2}=x$$
Why I alone isn't sufficient , since x/y <1/2 => x<y/2 .. As X is even less than half of Y so X can't be greater than Y
Please note some how statement 2 has been edited , part of statement 2 is missing, $$sqrt (y^2) = y$$, should be there.
The question as given now , gives answer as E.
Hope concerned persons will look into this.
Thank you.
Hi Stne,
We do not really need $$sqrt (y^2) = y$$ because if $$\sqrt{x^2}=x$$ is true, x is positive
and as x/y < 1/2 => 2x < y
Hence as y is greater than 2x, it has to be positive.
*press Kudos if you like the post!
You cannot cross multiply in inequality without knowing the signs of x and y , you can cross multiply only when you know that the signs are positive
if the question does not have $$sqrt (y^2) = y$$ then y need not necessarily be positive
consider x=1 and y = 3 this satisfies both the statement and the answer is no
consider x= 1 and y = -3 this satisfies both the statements and the answer is yes
as you can see, Y has to be positive for the answer to be c .
Hence question should be corrected, as in the present format, answer is E
Hope it helps , let me know if there is anything still unclear.[/quote]
But, incase y is negative:
x- y will always be greater than 0, given x is positive and x & y are non-zero integers.
For example 1- (-3) > 0
*press Kudos if you like the post!
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Re: If x and y are integers and xy 0, is x - y > 0 ? [#permalink]
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10 Apr 2014, 10:59
Hi Ricsingh,
x= 1 and y = 3 what is x-y ? its -2 so is x- y >0 answer No.
x=1 and y = -3 what is x- y ? its 4 so is x-y >0 answer yes
So what is the confusion? we have two different answers if $$sqrt(y^2)= y$$ is not given.
So it is very important to mention that Y is positive.
Hope this will help, if any thing is still unclear .let me know
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Re: If x and y are integers and xy 0, is x - y > 0 ? [#permalink]
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10 Apr 2014, 13:04
stne wrote:
Hi Ricsingh,
x= 1 and y = 3 what is x-y ? its -2 so is x- y >0 answer No.
x=1 and y = -3 what is x- y ? its 4 so is x-y >0 answer yes
So what is the confusion? we have two different answers if $$sqrt(y^2)= y$$ is not given.
So it is very important to mention that Y is positive.
Hope this will help, if any thing is still unclear .let me know
Hello - I agree that we need to know Y is positive or not for this question and I think someone has corrected it as well.
Just a thought, if we know Y is positive we do not really need to know what X is to answer the question.
because when x is negative x-y > 0 , not true
when x is positive, x-y >0 , not true if x is less than y and first statement confirms that.
Letme know your thoughts on this.
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Re: If x and y are integers and xy 0, is x - y > 0 ? [#permalink]
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11 Apr 2014, 02:44
shrive555 wrote:
If x and y are integers and xy ≠ 0, is x - y > 0 ?
(1) x/y < 1/2
(2) $$\sqrt{x^2}=x$$ and $$\sqrt{y^2}=y$$
[Reveal] Spoiler:
Why I alone isn't sufficient , since x/y <1/2 => x<y/2 .. As X is even less than half of Y so X can't be greater than Y
_______________________
Edited the second statement.
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Re: If x and y are integers and xy 0, is x - y > 0 ? [#permalink] 11 Apr 2014, 02:44
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# 2018 AMC 12B Problems/Problem 6
## Problem
Suppose $S$ cans of soda can be purchased from a vending machine for $Q$ quarters. Which of the following expressions describes the number of cans of soda that can be purchased for $D$ dollars, where 1 dollar is worth 4 quarters?
$\textbf{(A)} \frac{4DQ}{S} \qquad \textbf{(B)} \frac{4DS}{Q} \qquad \textbf{(C)} \frac{4Q}{DS} \qquad \textbf{(D)} \frac{DQ}{4S} \qquad \textbf{(E)} \frac{DS}{4Q}$
## Solution 1
The unit price for a can of soda (in quarters) is $\frac{S}{Q}$. Thus, the number of cans which can be bought for $D$ dollars ($4D$ quarters) is$\boxed {\textbf{(B)} \frac{4DS}{Q}}$ (Giraffefun)
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diagonal matrix python
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A Computer Science portal for geeks. With the help of Numpy matrix.diagonal() method, we are able to find a diagonal element from a given matrix and gives output as one dimensional matrix.. Syntax : matrix.diagonal() Return : Return diagonal element of a matrix Example #1 : In this example we can see that with the help of matrix.diagonal() method we are able to find the elements in a diagonal of a matrix. Parameters N int. The default is 0. If v is a 1-D array, return a 2-D array with v on the k-th diagonal. Number of rows in the output. Required: k: Diagonal in question. If a is 2-D and not a matrix, a 1-D array of the same type as a containing the diagonal is returned. This function modifies the input array in … optional The second printed matrix below it is v, whose columns are the eigenvectors corresponding to the eigenvalues in w. Meaning, to the w[i] eigenvalue, the corresponding eigenvector is the v[:,i] column in matrix v. In NumPy, the i th column vector of a matrix v is extracted as v[:,i] So, the eigenvalue w[0] goes with v[:,0] w[1] goes with v[:,1] Use k>0 for diagonals above the main diagonal, and k<0 for diagonals below the main diagonal. python3 app.py Main Diagonal: [1 5 7] Above main diagonal: [2 6] Below main diagonal: [ 4 8 15] In this example, we passed a 4×4 matrix and got the required output of the main diagonal, above the main diagonal ( k=1) and below the main diagonal(k=-1). License.All 697 notes and articles are available on GitHub.GitHub. If v is a 2-D array, return a copy of its k-th diagonal. Notes. Number of columns in the output. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview Questions. Welcome to part 12 of the Python 3 basics series, where we're going to be talking about how we can validate the winners of our TicTacToe game in the final way: diagonally! Return a 2-D array with ones on the diagonal and zeros elsewhere. If a is a matrix, a 1-D array containing the diagonal is returned in order to maintain backward compatibility. If None, defaults to N. k int, optional. My question is simply, what is an efficient way of calculating the degree of connectivity matrix, or is there a python module for that? If all the input arrays are square, the output is known as a block diagonal matrix. A square matrix is said to be scalar matrix if all the main diagonal elements are equal and other elements except main diagonal are zero. How to get the diagonal of a matrix in Python. M int, optional. Scalar matrix can also be written in form of n * I, where n is any real number and I is the identity matrix. numpy.fill_diagonal¶ numpy.fill_diagonal (a, val, wrap=False) [source] ¶ Fill the main diagonal of the given array of any dimensionality. Use np arange() function to create an array and then construct the diagonal. As we've done before, let's just start clean with a game example of a diagonal win: game = [[1, 0, 1], [1, 1, 2], [2, 2, 1]] For an array a with a.ndim >= 2, the diagonal is the list of locations with indices a[i,..., i] all identical.
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# The Gini Index
Document Sample
``` The Gini Index
Paige Stillwell
and
Tanya Picinich
Overview
• Lorenz Curve
– Perfect Income Equality / Complete Income Inequality
• Gini Index
– Calculation Examples
• United States Gini Index
• Riemann Sum and Trapezoidal Rule
• Potential Issues with the Gini Index
• Why is the Gini Index Important?
• Gini Index Comparison Across Countries
• United States Gini Index Over Time
The Lorenz Curve
• Shows the share of total income of the population
from 0 to t where t is the rank of a household’s income
as a percentage of the total population
Lorenz Curve for 2007 United States
1.2
1
Income Quintiles
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1 1.2
t values
Perfect Income Equality: Utopia
• 20% of the population makes 20% of the income and so on
• Lorenz curve has the equation y=x
Perfect Income Equality
1.2
1
Income Quintiles
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1 1.2
t values
Complete Inequality
• One person makes all the money. Everyone else
makes nothing
Total Inequality of Income
1.2
1
Income Quintiles
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1 1.2
t values
Gini Index
• Gives information about the income inequality of a country in
one number
• Ranges from 0 to 1
• Calculated as the area between perfect equality (y=x) and the
Lorenz curve
1
G(t ) 2 * (t Lorenz (t )) dt
0
Perfect Income Equality
1.2
1
• The Gini Index for
Income Quintiles
0.8
0.6
0.4
perfect equality is 0
0.2
0
0 0.2 0.4 0.6 0.8 1 1.2
t values
• The Gini Index for
total inequality is 1
Example: L(t) = t2
• What’s the Gini Index for a country that has a
Lorenz curve of L(t) = t2 ?
1
G 2 * (t t 2 )dt
0
t 2 t3 1
G 2*
2 3 0
1 1
G 2* 0
2 3
1
G .333
3
Example: L(t) = t3
• Will the Gini Index increase or decrease from the
previous example if the equation changes to L(t) = t3?
1
G 2 * (t t 3 )dt
0
t 2 t4 1
G 2*
2 4 0
1 1
G 2* 0
2 4
1
G .50
2
Reality: How the Gini Index is
Calculated
• In real life we are not given functions
• We must use data points to find the Gini Index
• We use income quintiles which are made available by
the U.S. Census Bureau
United States Income Quintiles 2007
Income Lower Second Middle Fourth Highest
Quintiles 5th 5th 5th 5th 5th
% of
total 55.3 25.9 11.3 4.4 3.1
income
The United States Gini Index 2007
• The United States Gini
Index for 2007 is quoted
by the U.S. Census
Bureau as G(t)=.463
• This value may vary
depending on what is
considered as income
and whether individuals
or households are
examined
Riemann Sum and
Trapezoidal Rule
• Approximates an integral when the equation of the function is
unknown
• Using the 2007 data we can approximate the Gini Index
United States Income Quintiles 2007
t 0 .20 .40 .60 .80 1
L(t) 0 .553 .812 .925 .969 1
• Riemann sums use either right or left endpoints to form rectangles
• The trapezoidal rule is the average of the right and left endpoint
approximations
Riemann Sum
• The definition of an integral of f from a to b is:
f ( x)dx lim f x * x
a n
*
i
b n i 1
• Using right end points: • Using left end points:
xi* xi xi* xi 1
f ( x )dx lim f x * x f ( x )dx lim f x * x
a a n
n
i i 1
n i 1 b n i 1
b
x f ( x1 ) f ( x 2 ) f ( x3 ) f ( x4 ) f ( x5 ) x f ( x 0 ) f ( x1 ) f ( x2 ) f ( x3 ) f ( x 4 )
Riemann Sum
• Using right end points: •Using left end points:
R5
1
.553 .812 .925 .969 1 L5
1
0 .553 .812 .925 .969
5 5
1 1
.8518 - .6518 -
2 2
.3518 .1518
1 1
G 2 * R 5 G 2 * L5
2 2
G .7036 G .3036
Trapezoidal Rule
• Basically an average of the right and left endpoints from
the Riemann Sum
x
b
f ( x )dx * f ( x0 ) 2 f ( x1 ) 2 f ( x2 ) ... 2 f ( xn 1 ) f ( xn )
a
2
ba
where x and xi a ix
n
• Using the 2007 Income Quintiles as data points we get:
T5
.2
0 2(.553) 2(.812) 2(.925) 2(.969) 1
2
T5 .7518
Trapezoidal Rule
1 1
T5 - .7518 -
2 2
.2518
G 2 * Area
G 2 * .2518
G .5036
Actual Gini Index = .463
Potential Issues
• The Gini Index glosses over many details
• Gives a more accurate picture of the relationship
between the upper class and middle class than
the relationship between the upper class and
lower class
• Does not reflect unreported income and money
Ways to calculate a more accurate
Gini Index:
• Jackknife • Bootstrap
– Calculate the Gini Index – Calculate the Gini Index
many times, but remove from a random sample of
one data point each time the income data many
times
– Produces a mean
distribution and a – Produces a mean
standard deviation for the distribution and a
Gini Index standard deviation for the
Gini Index
Why is the Gini Index so important?
• Compiles information about income inequality
into 1 number
• Allows for comparisons with other countries
• Shows how income inequality changes over time
• This information has great social, political, and
economic implications
Comparisons Across Countries
Countries With Low Gini Indices Countries With High Gini Indices
Namibia 70.7
Sweden 23
Lesotho 63.2
Denmark 24 Central African Republic 61.3
Finland 26 Haiti 59.2
Bolivia 59.2
Brazil 56.7
• The Gini index for the entire world = 56 - 66
• Tolerance for inequality of income varies between countries
– US = higher tolerance for income inequality
– European countries = lower tolerance for income inequality
– Underdeveloped countries have a higher income inequality
United States Over Time
United States Gini Index over Time
50
40
Gini
30
20
10
0
1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020
Year
Questions?
```
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Please Note: This article is written for users of the following Microsoft Excel versions: 2007, 2010, 2013, 2016, 2019, and Excel in Microsoft 365. If you are using an earlier version (Excel 2003 or earlier), this tip may not work for you. For a version of this tip written specifically for earlier versions of Excel, click here: Averaging the Last Numbers in a Column.
# Averaging the Last Numbers in a Column
Written by Allen Wyatt (last updated July 10, 2021)
This tip applies to Excel 2007, 2010, 2013, 2016, 2019, and Excel in Microsoft 365
Emma has a list of numbers in a worksheet (let's say in column A) that are added to on a weekly basis. She needs to calculate the average of the last 12 numbers in the column. She wonders how she can do this and have the average always reflect the last 12 numbers, even when she keeps adding numbers each week.
Assuming that there are no gaps in your range of numbers, you can calculate the average of the last 12 numbers with this formula:
```=AVERAGE(OFFSET(A1,COUNTA(A:A)-12,0,12,1))
```
This formula should, of course, be placed in some cell that is not in column A. It uses the COUNTA function to figure out how many cells contain something in column A. If there are 100 cells in use in column A, this means that you end up with a formula being evaluated in this way:
```=AVERAGE(OFFSET(A1,100-12,0,12,1))
```
Of course, 100 minus 12 is 88, and this number is used as an offset from the starting cell (A2) to say that the range to be averaged should start at A89 and extend down 12 cells. That means that the average ends up being for the range A89:A100. As more numbers are added at the bottom of column A, the formula always reflects the last 12 numbers.
The formula will return an error if column A has fewer than 12 rows worth of data in it. To accommodate that possibility, you may want to alter the formula just a bit:
```=AVERAGE(OFFSET(A1,COUNTA(A:A)-MIN(COUNTA(A:A),12),0,MIN(COUNTA(A:A),12),1))
```
Instead of using a hard-and-fast value of 12 rows, the MIN function (in two places) returns the minimum of either the actual number of rows or 12. So, if your worksheet only has numbers in cells A1:A5, the MIN function would ensure that the formula only averaged those 5 values.
ExcelTips is your source for cost-effective Microsoft Excel training. This tip (10278) applies to Microsoft Excel 2007, 2010, 2013, 2016, 2019, and Excel in Microsoft 365. You can find a version of this tip for the older menu interface of Excel here: Averaging the Last Numbers in a Column.
##### Author Bio
Allen Wyatt
With more than 50 non-fiction books and numerous magazine articles to his credit, Allen Wyatt is an internationally recognized author. He is president of Sharon Parq Associates, a computer and publishing services company. ...
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# 80% Of 60
by -2 views
60 of 8000 480000. By 2050 80 of all older people will live in low- and middle-income countries.
Pin On Fonts Discounts
### 60 of 8050 483000.
80% of 60. Solution for 60 is what percent of 80 100×8060 100xx 8060x – we multiply both sides of the equation by x 100133333333333x – we divide both sides of the equation by 133333333333 to get x. Solution for what is 80 of 60 60×10080 60xx 10080x – we multiply both sides of the equation by x 60125x – we divide both sides of the equation by 125 to get x. If you want to use a calculator to know what is 80 percent of 60 simply enter 80 100 60 and you will get your answer which is 48 You may also be interested in.
What is 80 percent of 60. 12 is 20 of what. 80 of 6000 480000.
80 of 6001 480080. 80 of 6051 484080. This is the most common method to calculate 80 of 60.
STEP 1 60 80 Y STEP 2 60 80 100 Y Multiplying both sides by 100 and dividing both sides of the equation by 80 we will arrive at. The difference between 50 and 40 is divided by 40 and multiplied by 100. Below is the math and answer to What is 80 of 60 using the percent formula.
60 of 8026 481560. France had almost 150 years to adapt to a change from 10 to 20 in the proportion of the population that was older than 60 years. 60 of 8075 484500.
Whole Percent100 Part. The pace of population ageing around the world is also increasing dramatically. 60 is the Whole 80 is the Percent and the Part is what we are calculating.
2 How much to pay for an item of 60 when discounted 80 percent. In other words a 80 discount for a item with original price of 60 is equal to 48 Amount Saved. You can also calculate how much you save by simply moving the period in 8000 percent two spaces to the left and then multiply the result by 60 as follows.
80 of 6025 482000. 80 x 60 4800. 80 x 60100 4800 Final Price.
12 is divided by 20. 80 of 6075 486000. What is 80 off 60 Dollars An item that costs 60 when discounted 80 percent will cost 12 The easiest way of calculating discount is in this case to multiply the normal price 60 by 80 then divide it by one hundred.
60 of 8051 483060. 60 – 4800 1200 Thus a product that normally costs 60 with a 80 percent discount will cost you 1200 and you saved 4800. 60 of 8025 481500.
How much is 80 of 60. Steps to solve 60 is 80 percent of what number We have 80 x 60 or 80 100 x 60 Multiplying both sides by 100 and dividing both sides by 80 we have x 60 100 80 x 75. 60 x80 4800 savings.
You can also calculate how much you save by simply moving the period in 6000 percent two spaces to the left and then multiply the result by 80 as follows. 12 60 100 20. 80 of 6050 484000.
60 is 80 of 75. Use this easy and mobile-friendly calculator to calculate percentages. 80 560 An item that costs 56 when discounted 80 percent will cost 112 The easiest way of calculating discount is in this case to multiply the normal price 56 by 80 then divide it by one hundred.
In calculating 60 of a number sales tax credit cards cash back bonus interest discounts interest per annum dollars pounds coupons60 off 60 of price or something we use the formula above to find the answer. What is items sale price. 12 20 12 20 100 60.
Note that to find the amount saved just multiply it by the percentage and divide by 100. So the discount is equal to 448. 80 – 4800 3200 Thus a product that normally costs 80 with a 60 percent discount will cost you 3200 and you saved 4800.
What percent of 60 is 48 48 is what percent of 60. This calculator formula step by step calculation and associated information to find 80 of 60 80 percent of what number is 48 may help students teachers parents or professionals to learn teach practice or verify such percentage X of Y calculations efficiently. 60 of 8001 480060.
So the discount is equal to 48. 80 dollar to pound 528 pound How to calculate 60 off 80 dollars or pounds. What is the percentage change from 40 to 50.
If you are using a calculator simply enter 6010080 which will give you the answer. 80 of 6026 482080.
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# In the Given Fig, ∠ R is the Right Angle of δ Pqr. Write the Following Ratios. (I) Sin P (Ii) Cos Q (Iii) Tan P (Iv) Tan Q - Geometry
Sum
In the given Fig, angle R is the right angle of triangle PQR. Write the following ratios.
(i) sin P (ii) cos Q (iii) tan P (iv) tan Q
#### Solution
(i) Sin P = " Opposite side of ∠P"/" Hypotenuse " = ["QR"]/["PQ"]
(ii) CosQ = "Adjacent side of ∠Q"/" Hypotenuse " = ["QR"]/["PQ"]
(iii) tan P = " Opposite side of ∠P"/"Adjacent side of ∠P" = ["QR"]/["PR"]
(iv) tan Q = " Opposite side of ∠P"/"Adjacent side of ∠P" = ["PR "]/["QR"]
Is there an error in this question or solution?
#### APPEARS IN
Balbharati Mathematics 2 Geometry 9th Standard Maharashtra State Board
Chapter 8 Trigonometry
Practice Set 8.1 | Q 1 | Page 104
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# Common elements in all rows of a given matrix
• Difficulty Level : Medium
• Last Updated : 30 Jun, 2022
Given an m x n matrix, find all common elements present in all rows in O(mn) time and one traversal of matrix.
Example:
```Input:
mat[4][5] = {{1, 2, 1, 4, 8},
{3, 7, 8, 5, 1},
{8, 7, 7, 3, 1},
{8, 1, 2, 7, 9},
};
Output:
1 8 or 8 1
8 and 1 are present in all rows.```
A simple solution is to consider every element and check if it is present in all rows. If present, then print it.
A better solution is to sort all rows in the matrix and use similar approach as discussed here. Sorting will take O(mnlogn) time and finding common elements will take O(mn) time. So overall time complexity of this solution is O(mnlogn)
Can we do better than O(mnlogn)?
The idea is to use maps. We initially insert all elements of the first row in an map. For every other element in remaining rows, we check if it is present in the map. If it is present in the map and is not duplicated in current row, we increment count of the element in map by 1, else we ignore the element. If the currently traversed row is the last row, we print the element if it has appeared m-1 times before.
Below is the implementation of the idea:
## C++
`// A Program to prints common element in all``// rows of matrix``#include ``using` `namespace` `std;` `// Specify number of rows and columns``#define M 4``#define N 5` `// prints common element in all rows of matrix``void` `printCommonElements(``int` `mat[M][N])``{`` ``unordered_map<``int``, ``int``> mp;` ` ``// initialize 1st row elements with value 1`` ``for` `(``int` `j = 0; j < N; j++)`` ``mp[mat[0][j]] = 1;` ` ``// traverse the matrix`` ``for` `(``int` `i = 1; i < M; i++)`` ``{`` ``for` `(``int` `j = 0; j < N; j++)`` ``{`` ``// If element is present in the map and`` ``// is not duplicated in current row.`` ``if` `(mp[mat[i][j]] == i)`` ``{`` ``// we increment count of the element`` ``// in map by 1`` ``mp[mat[i][j]] = i + 1;` ` ``// If this is last row`` ``if` `(i==M-1 && mp[mat[i][j]]==M)`` ``cout << mat[i][j] << ``" "``;`` ``}`` ``}`` ``}``}` `// driver program to test above function``int` `main()``{`` ``int` `mat[M][N] =`` ``{`` ``{1, 2, 1, 4, 8},`` ``{3, 7, 8, 5, 1},`` ``{8, 7, 7, 3, 1},`` ``{8, 1, 2, 7, 9},`` ``};` ` ``printCommonElements(mat);` ` ``return` `0;``}`
## Java
`// Java Program to prints common element in all``// rows of matrix``import` `java.util.*;` `class` `GFG``{` `// Specify number of rows and columns``static` `int` `M = ``4``;``static` `int` `N =``5``;` `// prints common element in all rows of matrix``static` `void` `printCommonElements(``int` `mat[][])``{` ` ``Map mp = ``new` `HashMap<>();`` ` ` ``// initialize 1st row elements with value 1`` ``for` `(``int` `j = ``0``; j < N; j++)`` ``mp.put(mat[``0``][j],``1``);`` ` ` ``// traverse the matrix`` ``for` `(``int` `i = ``1``; i < M; i++)`` ``{`` ``for` `(``int` `j = ``0``; j < N; j++)`` ``{`` ``// If element is present in the map and`` ``// is not duplicated in current row.`` ``if` `(mp.get(mat[i][j]) != ``null` `&& mp.get(mat[i][j]) == i)`` ``{`` ``// we increment count of the element`` ``// in map by 1`` ``mp.put(mat[i][j], i + ``1``);` ` ``// If this is last row`` ``if` `(i == M - ``1``)`` ``System.out.print(mat[i][j] + ``" "``);`` ``}`` ``}`` ``}``}` `// Driver code``public` `static` `void` `main(String[] args)``{`` ``int` `mat[][] =`` ``{`` ``{``1``, ``2``, ``1``, ``4``, ``8``},`` ``{``3``, ``7``, ``8``, ``5``, ``1``},`` ``{``8``, ``7``, ``7``, ``3``, ``1``},`` ``{``8``, ``1``, ``2``, ``7``, ``9``},`` ``};` ` ``printCommonElements(mat);``}``}` `// This code contributed by Rajput-Ji`
## Python3
`# A Program to prints common element``# in all rows of matrix` `# Specify number of rows and columns``M ``=` `4``N ``=` `5` `# prints common element in all``# rows of matrix``def` `printCommonElements(mat):` ` ``mp ``=` `dict``()` ` ``# initialize 1st row elements`` ``# with value 1`` ``for` `j ``in` `range``(N):`` ``mp[mat[``0``][j]] ``=` `1` ` ``# traverse the matrix`` ``for` `i ``in` `range``(``1``, M):`` ``for` `j ``in` `range``(N):`` ` ` ``# If element is present in the`` ``# map and is not duplicated in`` ``# current row.`` ``if` `(mat[i][j] ``in` `mp.keys() ``and`` ``mp[mat[i][j]] ``=``=` `i):`` ` ` ``# we increment count of the`` ``# element in map by 1`` ``mp[mat[i][j]] ``=` `i ``+` `1` ` ``# If this is last row`` ``if` `i ``=``=` `M ``-` `1``:`` ``print``(mat[i][j], end ``=` `" "``)`` ` `# Driver Code``mat ``=` `[[``1``, ``2``, ``1``, ``4``, ``8``],`` ``[``3``, ``7``, ``8``, ``5``, ``1``],`` ``[``8``, ``7``, ``7``, ``3``, ``1``],`` ``[``8``, ``1``, ``2``, ``7``, ``9``]]` `printCommonElements(mat)` `# This code is contributed``# by mohit kumar 29`
## C#
`// C# Program to print common element in all``// rows of matrix to another.``using` `System;``using` `System.Collections.Generic;` `class` `GFG``{` `// Specify number of rows and columns``static` `int` `M = 4;``static` `int` `N = 5;` `// prints common element in all rows of matrix``static` `void` `printCommonElements(``int` `[,]mat)``{` ` ``Dictionary<``int``, ``int``> mp = ``new` `Dictionary<``int``, ``int``>();`` ` ` ``// initialize 1st row elements with value 1`` ``for` `(``int` `j = 0; j < N; j++)`` ``{`` ``if``(!mp.ContainsKey(mat[0, j]))`` ``mp.Add(mat[0, j], 1);`` ``}`` ` ` ``// traverse the matrix`` ``for` `(``int` `i = 1; i < M; i++)`` ``{`` ``for` `(``int` `j = 0; j < N; j++)`` ``{`` ``// If element is present in the map and`` ``// is not duplicated in current row.`` ``if` `(mp.ContainsKey(mat[i, j])&&`` ``(mp[mat[i, j]] != 0 &&`` ``mp[mat[i, j]] == i))`` ``{`` ``// we increment count of the element`` ``// in map by 1`` ``if``(mp.ContainsKey(mat[i, j]))`` ``{`` ``var` `v = mp[mat[i, j]];`` ``mp.Remove(mat[i, j]);`` ``mp.Add(mat[i, j], i + 1);`` ``}`` ``else`` ``mp.Add(mat[i, j], i + 1);` ` ``// If this is last row`` ``if` `(i == M - 1)`` ``Console.Write(mat[i, j] + ``" "``);`` ``}`` ``}`` ``}``}` `// Driver code``public` `static` `void` `Main(String[] args)``{`` ``int` `[,]mat = {{1, 2, 1, 4, 8},`` ``{3, 7, 8, 5, 1},`` ``{8, 7, 7, 3, 1},`` ``{8, 1, 2, 7, 9}};` ` ``printCommonElements(mat);``}``}` `// This code is contributed by 29AjayKumar`
## Javascript
``
Output
`8 1 `
The time complexity of this solution is O(m * n) and we are doing only one traversal of the matrix.
Auxiliary Space: O(N)
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# Can also be measured in length as a portion of the
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can also be measured in length, as a portion of the circumference. Arc Length: The length of an arc or a portion of a circle’s circumference. The arc length is directly related to the degree arc measure. Example 5: Find the length of c PQ . Leave your answer in terms of π . Solution: In the picture, the central angle that corresponds with c PQ is 60 . This means that m c PQ = 60 . Think of the arc length as a portion of the circumference. There are 360 in a circle, so 60 would be 1 6 of that ( 60 360 = 1 6 ) . Therefore, the length of c PQ is 1 6 of the circumference. length of c PQ = 1 6 · 2 π (9) = 3 π Arc Length Formula: The length of c AB = m c AB 360 · π d or m c AB 360 · 2 π r . Another way to write this could be x 360 · 2 π r , where x is the central angle. Example 6: The arc length of c AB = 6 π and is 1 4 the circumference. Find the radius of the circle. Solution: If 6 π is 1 4 the circumference, then the total circumference is 4(6 π ) = 24 π . To find the radius, plug this into the circumference formula and solve for r . 24 π = 2 π r 12 = r Example 7: Find the measure of the central angle or c PQ . Solution: Let’s plug in what we know to the Arc Length Formula. 318
15 π = m c PQ 360 · 2 π (18) 15 = m c PQ 10 150 = m c PQ Example 8: The tires on a compact car are 18 inches in diameter. How far does the car travel after the tires turn once? How far does the car travel after 2500 rotations of the tires? Solution: One turn of the tire is the circumference. This would be C = 18 π 56 . 55 in . 2500 rotations would be 2500 · 56 . 55 in = 141371 . 67 in , 11781 ft, or 2.23 miles. Know What? Revisited The entire length of the crust, or the circumference of the pizza is 14 π 44 in . In 1 8 of the pizza, one piece would have 44 8 5 . 5 inches of crust. Review Questions • Questions 1-10 are similar to Examples 1 and 2. • Questions 11-14 are similar to Examples 3 and 4. • Questions 15-20 are similar to Example 5. • Questions 21-23 are similar to Example 6. • Questions 24-26 are similar to Example 7. • Questions 27-30 are similar to Example 8. Fill in the following table. Leave all answers in terms of π . Table 6.1: diameter radius circumference 1. 15 2. 4 3. 6 4. 84 π 5. 9 6. 25 π 7. 2 π 8. 36 9. Find the radius of circle with circumference 88 in. 319
10. Find the circumference of a circle with d = 20 π cm . Square PQS R is inscribed in T . RS = 8 2 . 11. Find the length of the diameter of T . 12. How does the diameter relate to PQS R ? 13. Find the perimeter of PQS R . 14. Find the circumference of T . Find the arc length of c PQ in A . Leave your answers in terms of π . 15. 16. 17. 18. 320
19. 20. Find PA (the radius) in A . Leave your answer in terms of π . 21. 22. 23. Find the central angle or m c PQ in A . Round any decimal answers to the nearest tenth. 24. 321
25. 26. For questions 27-30, a truck has tires with a 26 in diameter. 27. How far does the truck travel every time a tire turns exactly once? What is this the same as?
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## MATH
At Allstudybuddy our focus for early elementary classes is to boost their academic abilities and give them a head start in their academic journey. We introduce one math concept in several different ways, which allow the children to understand the math concept. It develops their thinking abilities. It may seem very easy to teach grade1, grade 2, and grade3. On the contrary, these are most crucial years for academics interest development. These little champs need perfect direction to be given in these initial years only. Every child is different. We choose the plan of study according to their mind.
It is important to consider, how finely kids outline the required knowledge and skills in detail and provide information and demonstrate their learning, how deeply they will explore concepts and at what level of complexity they will perform procedures, and the mathematical processes they will learn and apply throughout the grade.
## Number Sense and Numeration
Representing and ordering whole numbers to 50’s,100’s,and 1000’s; representing money amounts to 20¢ to \$10, decomposing and composing two-digit and three digit numbers establishing a one-to-one correspondence when counting the elements in a set; counting by 1’s, 2’s, 5’s, and 10’s, 25’s,and 100’s adding and subtracting up till three digit numbers,; investigating fractions of a adding and subtracting single digit to three-digit numbers in a variety of ways; relating one-digit multiplication, and division by one-digit divisors, and more.
## Measurement
Measuring using non-standard and standard units such as centimeters and meters and kilometers as telling time to the nearest half-hour, quarter-hour and to the nearest 5 minutes developing a sense of area; comparing mass and capacity of objects using measurable attributes; measuring length using; telling time to the nearest; measuring perimeter, area, mass, and capacity relating days to weeks and months to years relating minutes to hours, hours to days, days to weeks, and weeks to year
## Geometry and Spatial Sense
Classifying two-dimensional shapes and three dimensional figures symmetry; paths of motion Quadrilaterals congruent shapes recognizing transformations and more.
## Patterning and Algebra
Identifying, describing ,creating and extending growing and shrinking patterns; developing the concept of equality using the addition and subtraction using the commutative property and the property of zero ,a number line, and a bar graph; determining the missing numbers in equations involving addition and subtraction of one- and two-digit numbers.
## Data Management and Probability
Organizing objects into categories using one, two or more attributes; collecting and organizing categorical and discrete data; reading and displaying data using ,concrete graphs, pictographs vertical and horizontal bar graphs; describing probability, understanding mode
These are the final elementary academic years. This is utmost important that every student should have strong foundation of the basic concepts. Further classes are going to set direction for future career. These years of their age are very sensitive. They are open to learn new things and understand in a better manner, here it is must to talk to kids.
We at Allstudybuddy first start conversation with students. Talk about their understanding about core values, decision making, discipline and much more. It is of paramount importance to bring positivity in the attitude of a child.
• Numbers and Relations
• Operations with Decimal Numbers accordingly
• Multiplying and Dividing Fractions
• Ratios, Rates, and Proportions
• Powers and Exponents
• Volume
• Patterns and Solving Equations
• Statistics
• Tessellations
• Percents
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# Why does $∫{\frac{1+\log x}x}\,\mathrm dx$ not equal $\log x + \frac{(\log x)^2}2 + C?$ [closed]
When we assume that $$1+\log x = t,$$ then the integral becomes $$\log(1+\log x)+ C.$$
But when we assume that $$\log x = t,$$ the integral becomes $$∫\frac1x + \frac{\log x}x\,\mathrm dx \\ \log x + \frac{(\log x)^2}2 + C.$$ Why the discrepancy?
If you do the substitution $$t=1+\log{x}$$, the integral becomes$$\int t dt=\frac{1}{2}t^2+C=\frac{1}{2}(1+\log{x})^2+C=\frac{1}{2}(\log{x})^2+\log{x}+\frac{1}{2}+C$$ Combining $$\frac{1}{2}$$ and C into a new arbitrary constant gives you the same result
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# FormulaSheet (1) - 8 Effective Rate of Interest(1 1 k E i...
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Formula Sheet for the Final Exam 1. Circle : 2 2 2 ( ) ( ) x h y k r - + - = 2. Parabola (Quadratic function): 2 2 ( ) ( ) , ( ) or 2 4 b b y f x a x h k h k f h k c a a = = - + = - = = - 3. Distance formula: 2 2 2 1 2 1 ( ) ( ) d x x y y = - + - 4. Quadratic formula : 2 4 2 b b ac x a - ± - = For the following formulas : S is future value, P is present value, r is the annual interest rate, k is the number of compounding periods in a year, t is time in years, A is the amount of money, and R is the amount of payment; with the formula for the periodic interest rate r i k = . 5. Future Value of an Investment with continuously compounded interest: rt S Pe = (The amount at the end of an investment when an amount P is allowed to grow with interest compounded continuously.) 6. Future Value of an Investment: (1 ) kt S P i = + (The amount at the end of an investment when an amount P is allowed to grow.) 7. Present Value of an Investment: (1 ) kt P S i - = + (The amount that must be invested now to provide for a future value.)
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Unformatted text preview: 8. Effective Rate of Interest : (1 ) 1 k E i = +-(The effective rate for an account.) 9. Future Value of an Annuity: (1 ) 1 kt i S R i +-= (The amount at the end for an ordinary annuity with regular payments.) 10. Present Value of an Annuity: 1 (1 ) kt i P R i- - + = (The present value of an ordinary annuity with regular payments.) 11. ‘Sinking Fund’ Payment for an Annuity: (1 ) 1 kt Si R i = +-(The amount of a payment that will provide a future value of an ordinary annuity.) 12. Amortization Formula (Installment Payments): 1 (1 ) kt i R A i- = -+ (The amount of an installment payment when the amount borrowed is A .)...
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https://math.stackexchange.com/questions/3323231/proving-xn-y-has-2-solutions-if-n-even-natural-and-y0
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# Proving $x^n=y$ has 2 solutions if n even natural and $y>0$
Hi how is my proof for the question with $$n$$ an even natural number and $$y>0$$ $$x^n=y$$ has two solutions.
Assume n is some even natural number and $$y>0$$. The equation $$x^n=y$$ has one $$x>0$$ which satisfies it by the existence of roots.
Also $$x^n=(-x)^n$$ for all $$x$$. Therefore
$$x^n=y$$ if and only if $$(-x)^n=y$$
Thanks
• The last part is fine. What exactly do you mean by existence of roots? x^2+1 has no root in $\mathbb{R}$, so what about the equation in question are you using? – WoolierThanThou Aug 14 at 15:42
• You mean real-valued solutions? – Wuestenfux Aug 14 at 15:44
• Thanks I meant to say that I'm assuming the existence of a unique x>0 which satisfies x^n=y and I can do this because of an assumption about nth roots – Carlos Bacca Aug 14 at 15:51
• Yes real valued – Carlos Bacca Aug 14 at 15:52
## 1 Answer
Your proof needs a little touching-up in some places:
• You say there is one $$x>0$$ such that $$x^n=y$$. You also need to say this $$x$$ is unique, so that the desired conclusion can be reached.
• After showing $$x^n=y\iff(-x)^n=y$$, you need to state that since there is a unique positive $$x$$ satisfying the original equation, this "reflection identity" means there is also a unique negative $$x$$ satisfying it.
• The case of $$x=0$$ should also be dealt with, but that is easy. Combining positive, negative and zero $$x$$, you may conclude the desired result.
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0
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Obtain DTFT of rectangular pulse, $x(n)= \begin{cases}A, & 0 \leq n \leq L-1 \\ 0, & \text { otherwise }\end{cases}$
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Solution:
$X\left(e^{j \omega}\right)=\sum_{n=0}^{L-1} A e^{-j \omega n}\\$
$=A\left[\frac{1-e^{-j \omega L}}{1-e^{-j \omega}}\right]\\$
$\sum_{n=0}^{N-1} x^n=\frac{1-x^N}{1-x}\\$
$X\left(e^{j \omega}\right)=A\left[\frac{\left(e^{\frac{j \omega L}{2}}-e^{\frac{-j \omega L}{2}}\right) e^{\frac{-j \omega L}{2}}}{\left(e^{\frac{j \omega}{2}}-e^{-\frac{j \omega}{2}}\right) e^{\frac{-j \omega}{2}}}\right]\\$
$=A\left[\frac{2 j \sin \frac{\omega L}{2}}{2 j \sin \frac{\omega}{2}}\right] e^{-\frac{j \omega(L-1)}{2}}\\$
$=A e^{-\frac{j \omega(L-1)}{2}}\left[\frac{\sin \frac{\omega L}{2}}{\sin \frac{\omega}{2}}\right]\\$
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