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1 | 6518-6521 | He earns a
profit of Rs17 50 per package on nuts and Rs 7 00 per package on bolts How
many packages of each should be produced each day so as to maximise his
profit, if he operates his machines for at the most 12 hours a day |
1 | 6519-6522 | 50 per package on nuts and Rs 7 00 per package on bolts How
many packages of each should be produced each day so as to maximise his
profit, if he operates his machines for at the most 12 hours a day 5 |
1 | 6520-6523 | 00 per package on bolts How
many packages of each should be produced each day so as to maximise his
profit, if he operates his machines for at the most 12 hours a day 5 A factory manufactures two types of screws, A and B |
1 | 6521-6524 | How
many packages of each should be produced each day so as to maximise his
profit, if he operates his machines for at the most 12 hours a day 5 A factory manufactures two types of screws, A and B Each type of screw
requires the use of two machines, an automatic and a hand operated |
1 | 6522-6525 | 5 A factory manufactures two types of screws, A and B Each type of screw
requires the use of two machines, an automatic and a hand operated It takes
4 minutes on the automatic and 6 minutes on hand operated machines to
manufacture a package of screws A, while it takes 6 minutes on automatic and
3 minutes on the hand operated machines to manufacture a package of screws
B |
1 | 6523-6526 | A factory manufactures two types of screws, A and B Each type of screw
requires the use of two machines, an automatic and a hand operated It takes
4 minutes on the automatic and 6 minutes on hand operated machines to
manufacture a package of screws A, while it takes 6 minutes on automatic and
3 minutes on the hand operated machines to manufacture a package of screws
B Each machine is available for at the most 4 hours on any day |
1 | 6524-6527 | Each type of screw
requires the use of two machines, an automatic and a hand operated It takes
4 minutes on the automatic and 6 minutes on hand operated machines to
manufacture a package of screws A, while it takes 6 minutes on automatic and
3 minutes on the hand operated machines to manufacture a package of screws
B Each machine is available for at the most 4 hours on any day The manufacturer
can sell a package of screws A at a profit of Rs 7 and screws B at a profit of
Rs 10 |
1 | 6525-6528 | It takes
4 minutes on the automatic and 6 minutes on hand operated machines to
manufacture a package of screws A, while it takes 6 minutes on automatic and
3 minutes on the hand operated machines to manufacture a package of screws
B Each machine is available for at the most 4 hours on any day The manufacturer
can sell a package of screws A at a profit of Rs 7 and screws B at a profit of
Rs 10 Assuming that he can sell all the screws he manufactures, how many
packages of each type should the factory owner produce in a day in order to
maximise his profit |
1 | 6526-6529 | Each machine is available for at the most 4 hours on any day The manufacturer
can sell a package of screws A at a profit of Rs 7 and screws B at a profit of
Rs 10 Assuming that he can sell all the screws he manufactures, how many
packages of each type should the factory owner produce in a day in order to
maximise his profit Determine the maximum profit |
1 | 6527-6530 | The manufacturer
can sell a package of screws A at a profit of Rs 7 and screws B at a profit of
Rs 10 Assuming that he can sell all the screws he manufactures, how many
packages of each type should the factory owner produce in a day in order to
maximise his profit Determine the maximum profit © NCERT
not to be republished
520
MATHEMATICS
6 |
1 | 6528-6531 | Assuming that he can sell all the screws he manufactures, how many
packages of each type should the factory owner produce in a day in order to
maximise his profit Determine the maximum profit © NCERT
not to be republished
520
MATHEMATICS
6 A cottage industry manufactures pedestal lamps and wooden shades, each
requiring the use of a grinding/cutting machine and a sprayer |
1 | 6529-6532 | Determine the maximum profit © NCERT
not to be republished
520
MATHEMATICS
6 A cottage industry manufactures pedestal lamps and wooden shades, each
requiring the use of a grinding/cutting machine and a sprayer It takes 2 hours on
grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal
lamp |
1 | 6530-6533 | © NCERT
not to be republished
520
MATHEMATICS
6 A cottage industry manufactures pedestal lamps and wooden shades, each
requiring the use of a grinding/cutting machine and a sprayer It takes 2 hours on
grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal
lamp It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer
to manufacture a shade |
1 | 6531-6534 | A cottage industry manufactures pedestal lamps and wooden shades, each
requiring the use of a grinding/cutting machine and a sprayer It takes 2 hours on
grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal
lamp It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer
to manufacture a shade On any day, the sprayer is available for at the most 20
hours and the grinding/cutting machine for at the most 12 hours |
1 | 6532-6535 | It takes 2 hours on
grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal
lamp It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer
to manufacture a shade On any day, the sprayer is available for at the most 20
hours and the grinding/cutting machine for at the most 12 hours The profit from
the sale of a lamp is Rs 5 and that from a shade is Rs 3 |
1 | 6533-6536 | It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer
to manufacture a shade On any day, the sprayer is available for at the most 20
hours and the grinding/cutting machine for at the most 12 hours The profit from
the sale of a lamp is Rs 5 and that from a shade is Rs 3 Assuming that the
manufacturer can sell all the lamps and shades that he produces, how should he
schedule his daily production in order to maximise his profit |
1 | 6534-6537 | On any day, the sprayer is available for at the most 20
hours and the grinding/cutting machine for at the most 12 hours The profit from
the sale of a lamp is Rs 5 and that from a shade is Rs 3 Assuming that the
manufacturer can sell all the lamps and shades that he produces, how should he
schedule his daily production in order to maximise his profit 7 |
1 | 6535-6538 | The profit from
the sale of a lamp is Rs 5 and that from a shade is Rs 3 Assuming that the
manufacturer can sell all the lamps and shades that he produces, how should he
schedule his daily production in order to maximise his profit 7 A company manufactures two types of novelty souvenirs made of plywood |
1 | 6536-6539 | Assuming that the
manufacturer can sell all the lamps and shades that he produces, how should he
schedule his daily production in order to maximise his profit 7 A company manufactures two types of novelty souvenirs made of plywood Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for
assembling |
1 | 6537-6540 | 7 A company manufactures two types of novelty souvenirs made of plywood Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for
assembling Souvenirs of type B require 8 minutes each for cutting and 8 minutes
each for assembling |
1 | 6538-6541 | A company manufactures two types of novelty souvenirs made of plywood Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for
assembling Souvenirs of type B require 8 minutes each for cutting and 8 minutes
each for assembling There are 3 hours 20 minutes available for cutting and 4
hours for assembling |
1 | 6539-6542 | Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for
assembling Souvenirs of type B require 8 minutes each for cutting and 8 minutes
each for assembling There are 3 hours 20 minutes available for cutting and 4
hours for assembling The profit is Rs 5 each for type A and Rs 6 each for type
B souvenirs |
1 | 6540-6543 | Souvenirs of type B require 8 minutes each for cutting and 8 minutes
each for assembling There are 3 hours 20 minutes available for cutting and 4
hours for assembling The profit is Rs 5 each for type A and Rs 6 each for type
B souvenirs How many souvenirs of each type should the company manufacture
in order to maximise the profit |
1 | 6541-6544 | There are 3 hours 20 minutes available for cutting and 4
hours for assembling The profit is Rs 5 each for type A and Rs 6 each for type
B souvenirs How many souvenirs of each type should the company manufacture
in order to maximise the profit 8 |
1 | 6542-6545 | The profit is Rs 5 each for type A and Rs 6 each for type
B souvenirs How many souvenirs of each type should the company manufacture
in order to maximise the profit 8 A merchant plans to sell two types of personal computers – a desktop model and
a portable model that will cost Rs 25000 and Rs 40000 respectively |
1 | 6543-6546 | How many souvenirs of each type should the company manufacture
in order to maximise the profit 8 A merchant plans to sell two types of personal computers – a desktop model and
a portable model that will cost Rs 25000 and Rs 40000 respectively He estimates
that the total monthly demand of computers will not exceed 250 units |
1 | 6544-6547 | 8 A merchant plans to sell two types of personal computers – a desktop model and
a portable model that will cost Rs 25000 and Rs 40000 respectively He estimates
that the total monthly demand of computers will not exceed 250 units Determine
the number of units of each type of computers which the merchant should stock
to get maximum profit if he does not want to invest more than Rs 70 lakhs and if
his profit on the desktop model is Rs 4500 and on portable model is Rs 5000 |
1 | 6545-6548 | A merchant plans to sell two types of personal computers – a desktop model and
a portable model that will cost Rs 25000 and Rs 40000 respectively He estimates
that the total monthly demand of computers will not exceed 250 units Determine
the number of units of each type of computers which the merchant should stock
to get maximum profit if he does not want to invest more than Rs 70 lakhs and if
his profit on the desktop model is Rs 4500 and on portable model is Rs 5000 9 |
1 | 6546-6549 | He estimates
that the total monthly demand of computers will not exceed 250 units Determine
the number of units of each type of computers which the merchant should stock
to get maximum profit if he does not want to invest more than Rs 70 lakhs and if
his profit on the desktop model is Rs 4500 and on portable model is Rs 5000 9 A diet is to contain at least 80 units of vitamin A and 100 units of minerals |
1 | 6547-6550 | Determine
the number of units of each type of computers which the merchant should stock
to get maximum profit if he does not want to invest more than Rs 70 lakhs and if
his profit on the desktop model is Rs 4500 and on portable model is Rs 5000 9 A diet is to contain at least 80 units of vitamin A and 100 units of minerals Two
foods F1 and F2 are available |
1 | 6548-6551 | 9 A diet is to contain at least 80 units of vitamin A and 100 units of minerals Two
foods F1 and F2 are available Food F1 costs Rs 4 per unit food and F2 costs
Rs 6 per unit |
1 | 6549-6552 | A diet is to contain at least 80 units of vitamin A and 100 units of minerals Two
foods F1 and F2 are available Food F1 costs Rs 4 per unit food and F2 costs
Rs 6 per unit One unit of food F1 contains 3 units of vitamin A and 4 units of
minerals |
1 | 6550-6553 | Two
foods F1 and F2 are available Food F1 costs Rs 4 per unit food and F2 costs
Rs 6 per unit One unit of food F1 contains 3 units of vitamin A and 4 units of
minerals One unit of food F2 contains 6 units of vitamin A and 3 units of minerals |
1 | 6551-6554 | Food F1 costs Rs 4 per unit food and F2 costs
Rs 6 per unit One unit of food F1 contains 3 units of vitamin A and 4 units of
minerals One unit of food F2 contains 6 units of vitamin A and 3 units of minerals Formulate this as a linear programming problem |
1 | 6552-6555 | One unit of food F1 contains 3 units of vitamin A and 4 units of
minerals One unit of food F2 contains 6 units of vitamin A and 3 units of minerals Formulate this as a linear programming problem Find the minimum cost for diet
that consists of mixture of these two foods and also meets the minimal nutritional
requirements |
1 | 6553-6556 | One unit of food F2 contains 6 units of vitamin A and 3 units of minerals Formulate this as a linear programming problem Find the minimum cost for diet
that consists of mixture of these two foods and also meets the minimal nutritional
requirements 10 |
1 | 6554-6557 | Formulate this as a linear programming problem Find the minimum cost for diet
that consists of mixture of these two foods and also meets the minimal nutritional
requirements 10 There are two types of fertilisers F1 and F2 |
1 | 6555-6558 | Find the minimum cost for diet
that consists of mixture of these two foods and also meets the minimal nutritional
requirements 10 There are two types of fertilisers F1 and F2 F1 consists of 10% nitrogen and 6%
phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid |
1 | 6556-6559 | 10 There are two types of fertilisers F1 and F2 F1 consists of 10% nitrogen and 6%
phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid After
testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen
and 14 kg of phosphoric acid for her crop |
1 | 6557-6560 | There are two types of fertilisers F1 and F2 F1 consists of 10% nitrogen and 6%
phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid After
testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen
and 14 kg of phosphoric acid for her crop If F1 costs Rs 6/kg and F2 costs
Rs 5/kg, determine how much of each type of fertiliser should be used so that
nutrient requirements are met at a minimum cost |
1 | 6558-6561 | F1 consists of 10% nitrogen and 6%
phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid After
testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen
and 14 kg of phosphoric acid for her crop If F1 costs Rs 6/kg and F2 costs
Rs 5/kg, determine how much of each type of fertiliser should be used so that
nutrient requirements are met at a minimum cost What is the minimum cost |
1 | 6559-6562 | After
testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen
and 14 kg of phosphoric acid for her crop If F1 costs Rs 6/kg and F2 costs
Rs 5/kg, determine how much of each type of fertiliser should be used so that
nutrient requirements are met at a minimum cost What is the minimum cost 11 |
1 | 6560-6563 | If F1 costs Rs 6/kg and F2 costs
Rs 5/kg, determine how much of each type of fertiliser should be used so that
nutrient requirements are met at a minimum cost What is the minimum cost 11 The corner points of the feasible region determined by the following system of
linear inequalities:
2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5) |
1 | 6561-6564 | What is the minimum cost 11 The corner points of the feasible region determined by the following system of
linear inequalities:
2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5) Let
Z = px + qy, where p, q > 0 |
1 | 6562-6565 | 11 The corner points of the feasible region determined by the following system of
linear inequalities:
2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5) Let
Z = px + qy, where p, q > 0 Condition on p and q so that the maximum of Z
occurs at both (3, 4) and (0, 5) is
(A) p = q
(B) p = 2q
(C) p = 3q
(D) q = 3p
© NCERT
not to be republished
LINEAR PROGRAMMING 521
Miscellaneous Examples
Example 9 (Diet problem) A dietician has to develop a special diet using two foods
P and Q |
1 | 6563-6566 | The corner points of the feasible region determined by the following system of
linear inequalities:
2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5) Let
Z = px + qy, where p, q > 0 Condition on p and q so that the maximum of Z
occurs at both (3, 4) and (0, 5) is
(A) p = q
(B) p = 2q
(C) p = 3q
(D) q = 3p
© NCERT
not to be republished
LINEAR PROGRAMMING 521
Miscellaneous Examples
Example 9 (Diet problem) A dietician has to develop a special diet using two foods
P and Q Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units
of iron, 6 units of cholesterol and 6 units of vitamin A |
1 | 6564-6567 | Let
Z = px + qy, where p, q > 0 Condition on p and q so that the maximum of Z
occurs at both (3, 4) and (0, 5) is
(A) p = q
(B) p = 2q
(C) p = 3q
(D) q = 3p
© NCERT
not to be republished
LINEAR PROGRAMMING 521
Miscellaneous Examples
Example 9 (Diet problem) A dietician has to develop a special diet using two foods
P and Q Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units
of iron, 6 units of cholesterol and 6 units of vitamin A Each packet of the same quantity
of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units
of vitamin A |
1 | 6565-6568 | Condition on p and q so that the maximum of Z
occurs at both (3, 4) and (0, 5) is
(A) p = q
(B) p = 2q
(C) p = 3q
(D) q = 3p
© NCERT
not to be republished
LINEAR PROGRAMMING 521
Miscellaneous Examples
Example 9 (Diet problem) A dietician has to develop a special diet using two foods
P and Q Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units
of iron, 6 units of cholesterol and 6 units of vitamin A Each packet of the same quantity
of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units
of vitamin A The diet requires atleast 240 units of calcium, atleast 460 units of iron and
at most 300 units of cholesterol |
1 | 6566-6569 | Each packet (containing 30 g) of food P contains 12 units of calcium, 4 units
of iron, 6 units of cholesterol and 6 units of vitamin A Each packet of the same quantity
of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units
of vitamin A The diet requires atleast 240 units of calcium, atleast 460 units of iron and
at most 300 units of cholesterol How many packets of each food should be used to
minimise the amount of vitamin A in the diet |
1 | 6567-6570 | Each packet of the same quantity
of food Q contains 3 units of calcium, 20 units of iron, 4 units of cholesterol and 3 units
of vitamin A The diet requires atleast 240 units of calcium, atleast 460 units of iron and
at most 300 units of cholesterol How many packets of each food should be used to
minimise the amount of vitamin A in the diet What is the minimum amount of vitamin A |
1 | 6568-6571 | The diet requires atleast 240 units of calcium, atleast 460 units of iron and
at most 300 units of cholesterol How many packets of each food should be used to
minimise the amount of vitamin A in the diet What is the minimum amount of vitamin A Solution Let x and y be the number of packets of food P and Q respectively |
1 | 6569-6572 | How many packets of each food should be used to
minimise the amount of vitamin A in the diet What is the minimum amount of vitamin A Solution Let x and y be the number of packets of food P and Q respectively Obviously
x ≥ 0, y ≥ 0 |
1 | 6570-6573 | What is the minimum amount of vitamin A Solution Let x and y be the number of packets of food P and Q respectively Obviously
x ≥ 0, y ≥ 0 Mathematical formulation of the given problem is as follows:
Minimise Z = 6x + 3y (vitamin A)
subject to the constraints
12x + 3y ≥ 240 (constraint on calcium), i |
1 | 6571-6574 | Solution Let x and y be the number of packets of food P and Q respectively Obviously
x ≥ 0, y ≥ 0 Mathematical formulation of the given problem is as follows:
Minimise Z = 6x + 3y (vitamin A)
subject to the constraints
12x + 3y ≥ 240 (constraint on calcium), i e |
1 | 6572-6575 | Obviously
x ≥ 0, y ≥ 0 Mathematical formulation of the given problem is as follows:
Minimise Z = 6x + 3y (vitamin A)
subject to the constraints
12x + 3y ≥ 240 (constraint on calcium), i e 4x + y ≥ 80 |
1 | 6573-6576 | Mathematical formulation of the given problem is as follows:
Minimise Z = 6x + 3y (vitamin A)
subject to the constraints
12x + 3y ≥ 240 (constraint on calcium), i e 4x + y ≥ 80 (1)
4x + 20y ≥ 460 (constraint on iron), i |
1 | 6574-6577 | e 4x + y ≥ 80 (1)
4x + 20y ≥ 460 (constraint on iron), i e |
1 | 6575-6578 | 4x + y ≥ 80 (1)
4x + 20y ≥ 460 (constraint on iron), i e x + 5y ≥ 115 |
1 | 6576-6579 | (1)
4x + 20y ≥ 460 (constraint on iron), i e x + 5y ≥ 115 (2)
6x + 4y ≤ 300 (constraint on cholesterol), i |
1 | 6577-6580 | e x + 5y ≥ 115 (2)
6x + 4y ≤ 300 (constraint on cholesterol), i e |
1 | 6578-6581 | x + 5y ≥ 115 (2)
6x + 4y ≤ 300 (constraint on cholesterol), i e 3x + 2y ≤ 150 |
1 | 6579-6582 | (2)
6x + 4y ≤ 300 (constraint on cholesterol), i e 3x + 2y ≤ 150 (3)
x ≥ 0, y ≥ 0 |
1 | 6580-6583 | e 3x + 2y ≤ 150 (3)
x ≥ 0, y ≥ 0 (4)
Let us graph the inequalities (1) to (4) |
1 | 6581-6584 | 3x + 2y ≤ 150 (3)
x ≥ 0, y ≥ 0 (4)
Let us graph the inequalities (1) to (4) The feasible region (shaded) determined by the constraints (1) to (4) is shown in
Fig 12 |
1 | 6582-6585 | (3)
x ≥ 0, y ≥ 0 (4)
Let us graph the inequalities (1) to (4) The feasible region (shaded) determined by the constraints (1) to (4) is shown in
Fig 12 10 and note that it is bounded |
1 | 6583-6586 | (4)
Let us graph the inequalities (1) to (4) The feasible region (shaded) determined by the constraints (1) to (4) is shown in
Fig 12 10 and note that it is bounded Fig 12 |
1 | 6584-6587 | The feasible region (shaded) determined by the constraints (1) to (4) is shown in
Fig 12 10 and note that it is bounded Fig 12 10
© NCERT
not to be republished
522
MATHEMATICS
The coordinates of the corner points L, M and N are (2, 72), (15, 20) and (40, 15)
respectively |
1 | 6585-6588 | 10 and note that it is bounded Fig 12 10
© NCERT
not to be republished
522
MATHEMATICS
The coordinates of the corner points L, M and N are (2, 72), (15, 20) and (40, 15)
respectively Let us evaluate Z at these points:
Corner Point
Z = 6 x + 3 y
(2, 72)
228
(15, 20)
150 ←
Minimum
(40, 15)
285
From the table, we find that Z is minimum at the point (15, 20) |
1 | 6586-6589 | Fig 12 10
© NCERT
not to be republished
522
MATHEMATICS
The coordinates of the corner points L, M and N are (2, 72), (15, 20) and (40, 15)
respectively Let us evaluate Z at these points:
Corner Point
Z = 6 x + 3 y
(2, 72)
228
(15, 20)
150 ←
Minimum
(40, 15)
285
From the table, we find that Z is minimum at the point (15, 20) Hence, the amount
of vitamin A under the constraints given in the problem will be minimum, if 15 packets
of food P and 20 packets of food Q are used in the special diet |
1 | 6587-6590 | 10
© NCERT
not to be republished
522
MATHEMATICS
The coordinates of the corner points L, M and N are (2, 72), (15, 20) and (40, 15)
respectively Let us evaluate Z at these points:
Corner Point
Z = 6 x + 3 y
(2, 72)
228
(15, 20)
150 ←
Minimum
(40, 15)
285
From the table, we find that Z is minimum at the point (15, 20) Hence, the amount
of vitamin A under the constraints given in the problem will be minimum, if 15 packets
of food P and 20 packets of food Q are used in the special diet The minimum amount
of vitamin A will be 150 units |
1 | 6588-6591 | Let us evaluate Z at these points:
Corner Point
Z = 6 x + 3 y
(2, 72)
228
(15, 20)
150 ←
Minimum
(40, 15)
285
From the table, we find that Z is minimum at the point (15, 20) Hence, the amount
of vitamin A under the constraints given in the problem will be minimum, if 15 packets
of food P and 20 packets of food Q are used in the special diet The minimum amount
of vitamin A will be 150 units Example 10 (Manufacturing problem) A manufacturer has three machines I, II
and III installed in his factory |
1 | 6589-6592 | Hence, the amount
of vitamin A under the constraints given in the problem will be minimum, if 15 packets
of food P and 20 packets of food Q are used in the special diet The minimum amount
of vitamin A will be 150 units Example 10 (Manufacturing problem) A manufacturer has three machines I, II
and III installed in his factory Machines I and II are capable of being operated for
at most 12 hours whereas machine III must be operated for atleast 5 hours a day |
1 | 6590-6593 | The minimum amount
of vitamin A will be 150 units Example 10 (Manufacturing problem) A manufacturer has three machines I, II
and III installed in his factory Machines I and II are capable of being operated for
at most 12 hours whereas machine III must be operated for atleast 5 hours a day She
produces only two items M and N each requiring the use of all the three machines |
1 | 6591-6594 | Example 10 (Manufacturing problem) A manufacturer has three machines I, II
and III installed in his factory Machines I and II are capable of being operated for
at most 12 hours whereas machine III must be operated for atleast 5 hours a day She
produces only two items M and N each requiring the use of all the three machines The number of hours required for producing 1 unit of each of M and N on the three
machines are given in the following table:
Items
Number of hours required on machines
I
II
III
M
1
2
1
N
2
1
1 |
1 | 6592-6595 | Machines I and II are capable of being operated for
at most 12 hours whereas machine III must be operated for atleast 5 hours a day She
produces only two items M and N each requiring the use of all the three machines The number of hours required for producing 1 unit of each of M and N on the three
machines are given in the following table:
Items
Number of hours required on machines
I
II
III
M
1
2
1
N
2
1
1 25
She makes a profit of Rs 600 and Rs 400 on items M and N respectively |
1 | 6593-6596 | She
produces only two items M and N each requiring the use of all the three machines The number of hours required for producing 1 unit of each of M and N on the three
machines are given in the following table:
Items
Number of hours required on machines
I
II
III
M
1
2
1
N
2
1
1 25
She makes a profit of Rs 600 and Rs 400 on items M and N respectively How many
of each item should she produce so as to maximise her profit assuming that she can sell
all the items that she produced |
1 | 6594-6597 | The number of hours required for producing 1 unit of each of M and N on the three
machines are given in the following table:
Items
Number of hours required on machines
I
II
III
M
1
2
1
N
2
1
1 25
She makes a profit of Rs 600 and Rs 400 on items M and N respectively How many
of each item should she produce so as to maximise her profit assuming that she can sell
all the items that she produced What will be the maximum profit |
1 | 6595-6598 | 25
She makes a profit of Rs 600 and Rs 400 on items M and N respectively How many
of each item should she produce so as to maximise her profit assuming that she can sell
all the items that she produced What will be the maximum profit Solution Let x and y be the number of items M and N respectively |
1 | 6596-6599 | How many
of each item should she produce so as to maximise her profit assuming that she can sell
all the items that she produced What will be the maximum profit Solution Let x and y be the number of items M and N respectively Total profit on the production = Rs (600 x + 400 y)
Mathematical formulation of the given problem is as follows:
Maximise Z = 600 x + 400 y
subject to the constraints:
x + 2y ≤ 12 (constraint on Machine I) |
1 | 6597-6600 | What will be the maximum profit Solution Let x and y be the number of items M and N respectively Total profit on the production = Rs (600 x + 400 y)
Mathematical formulation of the given problem is as follows:
Maximise Z = 600 x + 400 y
subject to the constraints:
x + 2y ≤ 12 (constraint on Machine I) (1)
2x + y ≤ 12 (constraint on Machine II) |
1 | 6598-6601 | Solution Let x and y be the number of items M and N respectively Total profit on the production = Rs (600 x + 400 y)
Mathematical formulation of the given problem is as follows:
Maximise Z = 600 x + 400 y
subject to the constraints:
x + 2y ≤ 12 (constraint on Machine I) (1)
2x + y ≤ 12 (constraint on Machine II) (2)
x + 5
4 y ≥ 5 (constraint on Machine III) |
1 | 6599-6602 | Total profit on the production = Rs (600 x + 400 y)
Mathematical formulation of the given problem is as follows:
Maximise Z = 600 x + 400 y
subject to the constraints:
x + 2y ≤ 12 (constraint on Machine I) (1)
2x + y ≤ 12 (constraint on Machine II) (2)
x + 5
4 y ≥ 5 (constraint on Machine III) (3)
x ≥ 0, y ≥ 0 |
1 | 6600-6603 | (1)
2x + y ≤ 12 (constraint on Machine II) (2)
x + 5
4 y ≥ 5 (constraint on Machine III) (3)
x ≥ 0, y ≥ 0 (4)
© NCERT
not to be republished
LINEAR PROGRAMMING 523
Let us draw the graph of constraints (1) to (4) |
1 | 6601-6604 | (2)
x + 5
4 y ≥ 5 (constraint on Machine III) (3)
x ≥ 0, y ≥ 0 (4)
© NCERT
not to be republished
LINEAR PROGRAMMING 523
Let us draw the graph of constraints (1) to (4) ABCDE is the feasible region
(shaded) as shown in Fig 12 |
1 | 6602-6605 | (3)
x ≥ 0, y ≥ 0 (4)
© NCERT
not to be republished
LINEAR PROGRAMMING 523
Let us draw the graph of constraints (1) to (4) ABCDE is the feasible region
(shaded) as shown in Fig 12 11 determined by the constraints (1) to (4) |
1 | 6603-6606 | (4)
© NCERT
not to be republished
LINEAR PROGRAMMING 523
Let us draw the graph of constraints (1) to (4) ABCDE is the feasible region
(shaded) as shown in Fig 12 11 determined by the constraints (1) to (4) Observe that
the feasible region is bounded, coordinates of the corner points A, B, C, D and E are
(5, 0) (6, 0), (4, 4), (0, 6) and (0, 4) respectively |
1 | 6604-6607 | ABCDE is the feasible region
(shaded) as shown in Fig 12 11 determined by the constraints (1) to (4) Observe that
the feasible region is bounded, coordinates of the corner points A, B, C, D and E are
(5, 0) (6, 0), (4, 4), (0, 6) and (0, 4) respectively Fig 12 |
1 | 6605-6608 | 11 determined by the constraints (1) to (4) Observe that
the feasible region is bounded, coordinates of the corner points A, B, C, D and E are
(5, 0) (6, 0), (4, 4), (0, 6) and (0, 4) respectively Fig 12 11
Let us evaluate Z = 600 x + 400 y at these corner points |
1 | 6606-6609 | Observe that
the feasible region is bounded, coordinates of the corner points A, B, C, D and E are
(5, 0) (6, 0), (4, 4), (0, 6) and (0, 4) respectively Fig 12 11
Let us evaluate Z = 600 x + 400 y at these corner points Corner point
Z = 600 x + 400 y
(5, 0)
3000
(6, 0)
3600
(4, 4)
4000 ←
Maximum
(0, 6)
2400
(0, 4)
1600
We see that the point (4, 4) is giving the maximum value of Z |
1 | 6607-6610 | Fig 12 11
Let us evaluate Z = 600 x + 400 y at these corner points Corner point
Z = 600 x + 400 y
(5, 0)
3000
(6, 0)
3600
(4, 4)
4000 ←
Maximum
(0, 6)
2400
(0, 4)
1600
We see that the point (4, 4) is giving the maximum value of Z Hence, the
manufacturer has to produce 4 units of each item to get the maximum profit of Rs 4000 |
1 | 6608-6611 | 11
Let us evaluate Z = 600 x + 400 y at these corner points Corner point
Z = 600 x + 400 y
(5, 0)
3000
(6, 0)
3600
(4, 4)
4000 ←
Maximum
(0, 6)
2400
(0, 4)
1600
We see that the point (4, 4) is giving the maximum value of Z Hence, the
manufacturer has to produce 4 units of each item to get the maximum profit of Rs 4000 Example 11 (Transportation problem) There are two factories located one at
place P and the other at place Q |
1 | 6609-6612 | Corner point
Z = 600 x + 400 y
(5, 0)
3000
(6, 0)
3600
(4, 4)
4000 ←
Maximum
(0, 6)
2400
(0, 4)
1600
We see that the point (4, 4) is giving the maximum value of Z Hence, the
manufacturer has to produce 4 units of each item to get the maximum profit of Rs 4000 Example 11 (Transportation problem) There are two factories located one at
place P and the other at place Q From these locations, a certain commodity is to be
delivered to each of the three depots situated at A, B and C |
1 | 6610-6613 | Hence, the
manufacturer has to produce 4 units of each item to get the maximum profit of Rs 4000 Example 11 (Transportation problem) There are two factories located one at
place P and the other at place Q From these locations, a certain commodity is to be
delivered to each of the three depots situated at A, B and C The weekly requirements
of the depots are respectively 5, 5 and 4 units of the commodity while the production
capacity of the factories at P and Q are respectively 8 and 6 units |
1 | 6611-6614 | Example 11 (Transportation problem) There are two factories located one at
place P and the other at place Q From these locations, a certain commodity is to be
delivered to each of the three depots situated at A, B and C The weekly requirements
of the depots are respectively 5, 5 and 4 units of the commodity while the production
capacity of the factories at P and Q are respectively 8 and 6 units The cost of
© NCERT
not to be republished
524
MATHEMATICS
P
8 units
A
5 units
C
4 units
Q
6 units
Factory
Factory
Depot
Depot
B
5 units
Rs 1005 – y
Rs 120
Rs 100
Rs 150
Rs 160
6
[(5
) + (5
)]
x
y
–
–
–
Depot
y
Rs 100
8 –
x–
y
5
x
–
x
transportation per unit is given below:
From/To
Cost (in Rs)
A
B
C
P
160
100
150
Q
100
120
100
How many units should be transported from each factory to each depot in order that
the transportation cost is minimum |
1 | 6612-6615 | From these locations, a certain commodity is to be
delivered to each of the three depots situated at A, B and C The weekly requirements
of the depots are respectively 5, 5 and 4 units of the commodity while the production
capacity of the factories at P and Q are respectively 8 and 6 units The cost of
© NCERT
not to be republished
524
MATHEMATICS
P
8 units
A
5 units
C
4 units
Q
6 units
Factory
Factory
Depot
Depot
B
5 units
Rs 1005 – y
Rs 120
Rs 100
Rs 150
Rs 160
6
[(5
) + (5
)]
x
y
–
–
–
Depot
y
Rs 100
8 –
x–
y
5
x
–
x
transportation per unit is given below:
From/To
Cost (in Rs)
A
B
C
P
160
100
150
Q
100
120
100
How many units should be transported from each factory to each depot in order that
the transportation cost is minimum What will be the minimum transportation cost |
1 | 6613-6616 | The weekly requirements
of the depots are respectively 5, 5 and 4 units of the commodity while the production
capacity of the factories at P and Q are respectively 8 and 6 units The cost of
© NCERT
not to be republished
524
MATHEMATICS
P
8 units
A
5 units
C
4 units
Q
6 units
Factory
Factory
Depot
Depot
B
5 units
Rs 1005 – y
Rs 120
Rs 100
Rs 150
Rs 160
6
[(5
) + (5
)]
x
y
–
–
–
Depot
y
Rs 100
8 –
x–
y
5
x
–
x
transportation per unit is given below:
From/To
Cost (in Rs)
A
B
C
P
160
100
150
Q
100
120
100
How many units should be transported from each factory to each depot in order that
the transportation cost is minimum What will be the minimum transportation cost Solution The problem can be explained diagrammatically as follows (Fig 12 |
1 | 6614-6617 | The cost of
© NCERT
not to be republished
524
MATHEMATICS
P
8 units
A
5 units
C
4 units
Q
6 units
Factory
Factory
Depot
Depot
B
5 units
Rs 1005 – y
Rs 120
Rs 100
Rs 150
Rs 160
6
[(5
) + (5
)]
x
y
–
–
–
Depot
y
Rs 100
8 –
x–
y
5
x
–
x
transportation per unit is given below:
From/To
Cost (in Rs)
A
B
C
P
160
100
150
Q
100
120
100
How many units should be transported from each factory to each depot in order that
the transportation cost is minimum What will be the minimum transportation cost Solution The problem can be explained diagrammatically as follows (Fig 12 12):
Let x units and y units of the commodity be transported from the factory at P to
the depots at A and B respectively |
1 | 6615-6618 | What will be the minimum transportation cost Solution The problem can be explained diagrammatically as follows (Fig 12 12):
Let x units and y units of the commodity be transported from the factory at P to
the depots at A and B respectively Then (8 – x – y) units will be transported to depot
at C (Why |
1 | 6616-6619 | Solution The problem can be explained diagrammatically as follows (Fig 12 12):
Let x units and y units of the commodity be transported from the factory at P to
the depots at A and B respectively Then (8 – x – y) units will be transported to depot
at C (Why )
Hence, we have
x ≥ 0, y ≥ 0
and
8 – x – y ≥ 0
i |
1 | 6617-6620 | 12):
Let x units and y units of the commodity be transported from the factory at P to
the depots at A and B respectively Then (8 – x – y) units will be transported to depot
at C (Why )
Hence, we have
x ≥ 0, y ≥ 0
and
8 – x – y ≥ 0
i e |
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