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1 | 6418-6421 | 9 Maximise Z = – x + 2y, subject to the constraints:
x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0 10 Maximise Z = x + y, subject to x – y ≤ –1, –x + y ≤ 0, x, y ≥ 0 |
1 | 6419-6422 | Maximise Z = – x + 2y, subject to the constraints:
x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0 10 Maximise Z = x + y, subject to x – y ≤ –1, –x + y ≤ 0, x, y ≥ 0 12 |
1 | 6420-6423 | 10 Maximise Z = x + y, subject to x – y ≤ –1, –x + y ≤ 0, x, y ≥ 0 12 3 Different Types of Linear Programming Problems
A few important linear programming problems are listed below:
1 |
1 | 6421-6424 | Maximise Z = x + y, subject to x – y ≤ –1, –x + y ≤ 0, x, y ≥ 0 12 3 Different Types of Linear Programming Problems
A few important linear programming problems are listed below:
1 Manufacturing problems In these problems, we determine the number of units
of different products which should be produced and sold by a firm
when each product requires a fixed manpower, machine hours, labour hour per
unit of product, warehouse space per unit of the output etc |
1 | 6422-6425 | 12 3 Different Types of Linear Programming Problems
A few important linear programming problems are listed below:
1 Manufacturing problems In these problems, we determine the number of units
of different products which should be produced and sold by a firm
when each product requires a fixed manpower, machine hours, labour hour per
unit of product, warehouse space per unit of the output etc , in order to make
maximum profit |
1 | 6423-6426 | 3 Different Types of Linear Programming Problems
A few important linear programming problems are listed below:
1 Manufacturing problems In these problems, we determine the number of units
of different products which should be produced and sold by a firm
when each product requires a fixed manpower, machine hours, labour hour per
unit of product, warehouse space per unit of the output etc , in order to make
maximum profit 2 |
1 | 6424-6427 | Manufacturing problems In these problems, we determine the number of units
of different products which should be produced and sold by a firm
when each product requires a fixed manpower, machine hours, labour hour per
unit of product, warehouse space per unit of the output etc , in order to make
maximum profit 2 Diet problems In these problems, we determine the amount of different kinds
of constituents/nutrients which should be included in a diet so as to minimise the
cost of the desired diet such that it contains a certain minimum amount of each
constituent/nutrients |
1 | 6425-6428 | , in order to make
maximum profit 2 Diet problems In these problems, we determine the amount of different kinds
of constituents/nutrients which should be included in a diet so as to minimise the
cost of the desired diet such that it contains a certain minimum amount of each
constituent/nutrients 3 |
1 | 6426-6429 | 2 Diet problems In these problems, we determine the amount of different kinds
of constituents/nutrients which should be included in a diet so as to minimise the
cost of the desired diet such that it contains a certain minimum amount of each
constituent/nutrients 3 Transportation problems In these problems, we determine a transportation
schedule in order to find the cheapest way of transporting a product from
plants/factories situated at different locations to different markets |
1 | 6427-6430 | Diet problems In these problems, we determine the amount of different kinds
of constituents/nutrients which should be included in a diet so as to minimise the
cost of the desired diet such that it contains a certain minimum amount of each
constituent/nutrients 3 Transportation problems In these problems, we determine a transportation
schedule in order to find the cheapest way of transporting a product from
plants/factories situated at different locations to different markets © NCERT
not to be republished
LINEAR PROGRAMMING 515
Let us now solve some of these types of linear programming problems:
Example 6 (Diet problem): A dietician wishes to mix two types of foods in such a
way that vitamin contents of the mixture contain atleast 8 units of vitamin A and 10
units of vitamin C |
1 | 6428-6431 | 3 Transportation problems In these problems, we determine a transportation
schedule in order to find the cheapest way of transporting a product from
plants/factories situated at different locations to different markets © NCERT
not to be republished
LINEAR PROGRAMMING 515
Let us now solve some of these types of linear programming problems:
Example 6 (Diet problem): A dietician wishes to mix two types of foods in such a
way that vitamin contents of the mixture contain atleast 8 units of vitamin A and 10
units of vitamin C Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C |
1 | 6429-6432 | Transportation problems In these problems, we determine a transportation
schedule in order to find the cheapest way of transporting a product from
plants/factories situated at different locations to different markets © NCERT
not to be republished
LINEAR PROGRAMMING 515
Let us now solve some of these types of linear programming problems:
Example 6 (Diet problem): A dietician wishes to mix two types of foods in such a
way that vitamin contents of the mixture contain atleast 8 units of vitamin A and 10
units of vitamin C Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C Food ‘II’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C |
1 | 6430-6433 | © NCERT
not to be republished
LINEAR PROGRAMMING 515
Let us now solve some of these types of linear programming problems:
Example 6 (Diet problem): A dietician wishes to mix two types of foods in such a
way that vitamin contents of the mixture contain atleast 8 units of vitamin A and 10
units of vitamin C Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C Food ‘II’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C It costs
Rs 50 per kg to purchase Food ‘I’ and Rs 70 per kg to purchase Food ‘II’ |
1 | 6431-6434 | Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C Food ‘II’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C It costs
Rs 50 per kg to purchase Food ‘I’ and Rs 70 per kg to purchase Food ‘II’ Formulate
this problem as a linear programming problem to minimise the cost of such a mixture |
1 | 6432-6435 | Food ‘II’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C It costs
Rs 50 per kg to purchase Food ‘I’ and Rs 70 per kg to purchase Food ‘II’ Formulate
this problem as a linear programming problem to minimise the cost of such a mixture Solution Let the mixture contain x kg of Food ‘I’ and y kg of Food ‘II’ |
1 | 6433-6436 | It costs
Rs 50 per kg to purchase Food ‘I’ and Rs 70 per kg to purchase Food ‘II’ Formulate
this problem as a linear programming problem to minimise the cost of such a mixture Solution Let the mixture contain x kg of Food ‘I’ and y kg of Food ‘II’ Clearly, x ≥ 0,
y ≥ 0 |
1 | 6434-6437 | Formulate
this problem as a linear programming problem to minimise the cost of such a mixture Solution Let the mixture contain x kg of Food ‘I’ and y kg of Food ‘II’ Clearly, x ≥ 0,
y ≥ 0 We make the following table from the given data:
Resources
Food
Requirement
I
II
(x)
(y)
Vitamin A
2
1
8
(units/kg)
Vitamin C
1
2
10
(units/kg)
Cost (Rs/kg)
50
70
Since the mixture must contain at least 8 units of vitamin A and 10 units of
vitamin C, we have the constraints:
2x + y ≥ 8
x + 2y ≥ 10
Total cost Z of purchasing x kg of food ‘I’ and y kg of Food ‘II’ is
Z = 50x + 70y
Hence, the mathematical formulation of the problem is:
Minimise
Z = 50x + 70y |
1 | 6435-6438 | Solution Let the mixture contain x kg of Food ‘I’ and y kg of Food ‘II’ Clearly, x ≥ 0,
y ≥ 0 We make the following table from the given data:
Resources
Food
Requirement
I
II
(x)
(y)
Vitamin A
2
1
8
(units/kg)
Vitamin C
1
2
10
(units/kg)
Cost (Rs/kg)
50
70
Since the mixture must contain at least 8 units of vitamin A and 10 units of
vitamin C, we have the constraints:
2x + y ≥ 8
x + 2y ≥ 10
Total cost Z of purchasing x kg of food ‘I’ and y kg of Food ‘II’ is
Z = 50x + 70y
Hence, the mathematical formulation of the problem is:
Minimise
Z = 50x + 70y (1)
subject to the constraints:
2x + y ≥ 8 |
1 | 6436-6439 | Clearly, x ≥ 0,
y ≥ 0 We make the following table from the given data:
Resources
Food
Requirement
I
II
(x)
(y)
Vitamin A
2
1
8
(units/kg)
Vitamin C
1
2
10
(units/kg)
Cost (Rs/kg)
50
70
Since the mixture must contain at least 8 units of vitamin A and 10 units of
vitamin C, we have the constraints:
2x + y ≥ 8
x + 2y ≥ 10
Total cost Z of purchasing x kg of food ‘I’ and y kg of Food ‘II’ is
Z = 50x + 70y
Hence, the mathematical formulation of the problem is:
Minimise
Z = 50x + 70y (1)
subject to the constraints:
2x + y ≥ 8 (2)
x + 2y ≥ 10 |
1 | 6437-6440 | We make the following table from the given data:
Resources
Food
Requirement
I
II
(x)
(y)
Vitamin A
2
1
8
(units/kg)
Vitamin C
1
2
10
(units/kg)
Cost (Rs/kg)
50
70
Since the mixture must contain at least 8 units of vitamin A and 10 units of
vitamin C, we have the constraints:
2x + y ≥ 8
x + 2y ≥ 10
Total cost Z of purchasing x kg of food ‘I’ and y kg of Food ‘II’ is
Z = 50x + 70y
Hence, the mathematical formulation of the problem is:
Minimise
Z = 50x + 70y (1)
subject to the constraints:
2x + y ≥ 8 (2)
x + 2y ≥ 10 (3)
x, y ≥ 0 |
1 | 6438-6441 | (1)
subject to the constraints:
2x + y ≥ 8 (2)
x + 2y ≥ 10 (3)
x, y ≥ 0 (4)
Let us graph the inequalities (2) to (4) |
1 | 6439-6442 | (2)
x + 2y ≥ 10 (3)
x, y ≥ 0 (4)
Let us graph the inequalities (2) to (4) The feasible region determined by the
system is shown in the Fig 12 |
1 | 6440-6443 | (3)
x, y ≥ 0 (4)
Let us graph the inequalities (2) to (4) The feasible region determined by the
system is shown in the Fig 12 7 |
1 | 6441-6444 | (4)
Let us graph the inequalities (2) to (4) The feasible region determined by the
system is shown in the Fig 12 7 Here again, observe that the feasible region is
unbounded |
1 | 6442-6445 | The feasible region determined by the
system is shown in the Fig 12 7 Here again, observe that the feasible region is
unbounded © NCERT
not to be republished
516
MATHEMATICS
Let us evaluate Z at the corner points A(0,8), B(2,4) and C(10,0) |
1 | 6443-6446 | 7 Here again, observe that the feasible region is
unbounded © NCERT
not to be republished
516
MATHEMATICS
Let us evaluate Z at the corner points A(0,8), B(2,4) and C(10,0) Fig 12 |
1 | 6444-6447 | Here again, observe that the feasible region is
unbounded © NCERT
not to be republished
516
MATHEMATICS
Let us evaluate Z at the corner points A(0,8), B(2,4) and C(10,0) Fig 12 7
In the table, we find that smallest value of Z is 380 at the point (2,4) |
1 | 6445-6448 | © NCERT
not to be republished
516
MATHEMATICS
Let us evaluate Z at the corner points A(0,8), B(2,4) and C(10,0) Fig 12 7
In the table, we find that smallest value of Z is 380 at the point (2,4) Can we say
that the minimum value of Z is 380 |
1 | 6446-6449 | Fig 12 7
In the table, we find that smallest value of Z is 380 at the point (2,4) Can we say
that the minimum value of Z is 380 Remember that the feasible region is unbounded |
1 | 6447-6450 | 7
In the table, we find that smallest value of Z is 380 at the point (2,4) Can we say
that the minimum value of Z is 380 Remember that the feasible region is unbounded Therefore, we have to draw the graph of the inequality
50x + 70y < 380 i |
1 | 6448-6451 | Can we say
that the minimum value of Z is 380 Remember that the feasible region is unbounded Therefore, we have to draw the graph of the inequality
50x + 70y < 380 i e |
1 | 6449-6452 | Remember that the feasible region is unbounded Therefore, we have to draw the graph of the inequality
50x + 70y < 380 i e ,
5x + 7y < 38
to check whether the resulting open half plane has any point common with the feasible
region |
1 | 6450-6453 | Therefore, we have to draw the graph of the inequality
50x + 70y < 380 i e ,
5x + 7y < 38
to check whether the resulting open half plane has any point common with the feasible
region From the Fig 12 |
1 | 6451-6454 | e ,
5x + 7y < 38
to check whether the resulting open half plane has any point common with the feasible
region From the Fig 12 7, we see that it has no points in common |
1 | 6452-6455 | ,
5x + 7y < 38
to check whether the resulting open half plane has any point common with the feasible
region From the Fig 12 7, we see that it has no points in common Thus, the minimum value of Z is 380 attained at the point (2, 4) |
1 | 6453-6456 | From the Fig 12 7, we see that it has no points in common Thus, the minimum value of Z is 380 attained at the point (2, 4) Hence, the optimal
mixing strategy for the dietician would be to mix 2 kg of Food ‘I’ and 4 kg of Food ‘II’,
and with this strategy, the minimum cost of the mixture will be Rs 380 |
1 | 6454-6457 | 7, we see that it has no points in common Thus, the minimum value of Z is 380 attained at the point (2, 4) Hence, the optimal
mixing strategy for the dietician would be to mix 2 kg of Food ‘I’ and 4 kg of Food ‘II’,
and with this strategy, the minimum cost of the mixture will be Rs 380 Example 7 (Allocation problem) A cooperative society of farmers has 50 hectare
of land to grow two crops X and Y |
1 | 6455-6458 | Thus, the minimum value of Z is 380 attained at the point (2, 4) Hence, the optimal
mixing strategy for the dietician would be to mix 2 kg of Food ‘I’ and 4 kg of Food ‘II’,
and with this strategy, the minimum cost of the mixture will be Rs 380 Example 7 (Allocation problem) A cooperative society of farmers has 50 hectare
of land to grow two crops X and Y The profit from crops X and Y per hectare are
estimated as Rs 10,500 and Rs 9,000 respectively |
1 | 6456-6459 | Hence, the optimal
mixing strategy for the dietician would be to mix 2 kg of Food ‘I’ and 4 kg of Food ‘II’,
and with this strategy, the minimum cost of the mixture will be Rs 380 Example 7 (Allocation problem) A cooperative society of farmers has 50 hectare
of land to grow two crops X and Y The profit from crops X and Y per hectare are
estimated as Rs 10,500 and Rs 9,000 respectively To control weeds, a liquid herbicide
has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare |
1 | 6457-6460 | Example 7 (Allocation problem) A cooperative society of farmers has 50 hectare
of land to grow two crops X and Y The profit from crops X and Y per hectare are
estimated as Rs 10,500 and Rs 9,000 respectively To control weeds, a liquid herbicide
has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare Further,
no more than 800 litres of herbicide should be used in order to protect fish and wild life
using a pond which collects drainage from this land |
1 | 6458-6461 | The profit from crops X and Y per hectare are
estimated as Rs 10,500 and Rs 9,000 respectively To control weeds, a liquid herbicide
has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare Further,
no more than 800 litres of herbicide should be used in order to protect fish and wild life
using a pond which collects drainage from this land How much land should be allocated
to each crop so as to maximise the total profit of the society |
1 | 6459-6462 | To control weeds, a liquid herbicide
has to be used for crops X and Y at rates of 20 litres and 10 litres per hectare Further,
no more than 800 litres of herbicide should be used in order to protect fish and wild life
using a pond which collects drainage from this land How much land should be allocated
to each crop so as to maximise the total profit of the society Solution Let x hectare of land be allocated to crop X and y hectare to crop Y |
1 | 6460-6463 | Further,
no more than 800 litres of herbicide should be used in order to protect fish and wild life
using a pond which collects drainage from this land How much land should be allocated
to each crop so as to maximise the total profit of the society Solution Let x hectare of land be allocated to crop X and y hectare to crop Y Obviously,
x ≥ 0, y ≥ 0 |
1 | 6461-6464 | How much land should be allocated
to each crop so as to maximise the total profit of the society Solution Let x hectare of land be allocated to crop X and y hectare to crop Y Obviously,
x ≥ 0, y ≥ 0 Profit per hectare on crop X = Rs 10500
Profit per hectare on crop Y = Rs 9000
Therefore, total profit
= Rs (10500x + 9000y)
Corner Point
Z = 50x + 70y
(0,8)
560
(2,4)
380
(10,0)
500
Minimum
←
© NCERT
not to be republished
LINEAR PROGRAMMING 517
The mathematical formulation of the problem is as follows:
Maximise
Z = 10500 x + 9000 y
subject to the constraints:
x + y ≤ 50 (constraint related to land) |
1 | 6462-6465 | Solution Let x hectare of land be allocated to crop X and y hectare to crop Y Obviously,
x ≥ 0, y ≥ 0 Profit per hectare on crop X = Rs 10500
Profit per hectare on crop Y = Rs 9000
Therefore, total profit
= Rs (10500x + 9000y)
Corner Point
Z = 50x + 70y
(0,8)
560
(2,4)
380
(10,0)
500
Minimum
←
© NCERT
not to be republished
LINEAR PROGRAMMING 517
The mathematical formulation of the problem is as follows:
Maximise
Z = 10500 x + 9000 y
subject to the constraints:
x + y ≤ 50 (constraint related to land) (1)
20x + 10y ≤ 800 (constraint related to use of herbicide)
i |
1 | 6463-6466 | Obviously,
x ≥ 0, y ≥ 0 Profit per hectare on crop X = Rs 10500
Profit per hectare on crop Y = Rs 9000
Therefore, total profit
= Rs (10500x + 9000y)
Corner Point
Z = 50x + 70y
(0,8)
560
(2,4)
380
(10,0)
500
Minimum
←
© NCERT
not to be republished
LINEAR PROGRAMMING 517
The mathematical formulation of the problem is as follows:
Maximise
Z = 10500 x + 9000 y
subject to the constraints:
x + y ≤ 50 (constraint related to land) (1)
20x + 10y ≤ 800 (constraint related to use of herbicide)
i e |
1 | 6464-6467 | Profit per hectare on crop X = Rs 10500
Profit per hectare on crop Y = Rs 9000
Therefore, total profit
= Rs (10500x + 9000y)
Corner Point
Z = 50x + 70y
(0,8)
560
(2,4)
380
(10,0)
500
Minimum
←
© NCERT
not to be republished
LINEAR PROGRAMMING 517
The mathematical formulation of the problem is as follows:
Maximise
Z = 10500 x + 9000 y
subject to the constraints:
x + y ≤ 50 (constraint related to land) (1)
20x + 10y ≤ 800 (constraint related to use of herbicide)
i e 2x + y ≤ 80 |
1 | 6465-6468 | (1)
20x + 10y ≤ 800 (constraint related to use of herbicide)
i e 2x + y ≤ 80 (2)
x ≥ 0, y ≥ 0
(non negative constraint) |
1 | 6466-6469 | e 2x + y ≤ 80 (2)
x ≥ 0, y ≥ 0
(non negative constraint) (3)
Let us draw the graph of the system of inequalities (1) to (3) |
1 | 6467-6470 | 2x + y ≤ 80 (2)
x ≥ 0, y ≥ 0
(non negative constraint) (3)
Let us draw the graph of the system of inequalities (1) to (3) The feasible region
OABC is shown (shaded) in the Fig 12 |
1 | 6468-6471 | (2)
x ≥ 0, y ≥ 0
(non negative constraint) (3)
Let us draw the graph of the system of inequalities (1) to (3) The feasible region
OABC is shown (shaded) in the Fig 12 8 |
1 | 6469-6472 | (3)
Let us draw the graph of the system of inequalities (1) to (3) The feasible region
OABC is shown (shaded) in the Fig 12 8 Observe that the feasible region is bounded |
1 | 6470-6473 | The feasible region
OABC is shown (shaded) in the Fig 12 8 Observe that the feasible region is bounded The coordinates of the corner points O, A, B and C are (0, 0), (40, 0), (30, 20) and
(0, 50) respectively |
1 | 6471-6474 | 8 Observe that the feasible region is bounded The coordinates of the corner points O, A, B and C are (0, 0), (40, 0), (30, 20) and
(0, 50) respectively Let us evaluate the objective function Z = 10500 x + 9000y at
these vertices to find which one gives the maximum profit |
1 | 6472-6475 | Observe that the feasible region is bounded The coordinates of the corner points O, A, B and C are (0, 0), (40, 0), (30, 20) and
(0, 50) respectively Let us evaluate the objective function Z = 10500 x + 9000y at
these vertices to find which one gives the maximum profit Fig 12 |
1 | 6473-6476 | The coordinates of the corner points O, A, B and C are (0, 0), (40, 0), (30, 20) and
(0, 50) respectively Let us evaluate the objective function Z = 10500 x + 9000y at
these vertices to find which one gives the maximum profit Fig 12 8
Hence, the society will get the maximum profit of Rs 4,95,000 by allocating 30
hectares for crop X and 20 hectares for crop Y |
1 | 6474-6477 | Let us evaluate the objective function Z = 10500 x + 9000y at
these vertices to find which one gives the maximum profit Fig 12 8
Hence, the society will get the maximum profit of Rs 4,95,000 by allocating 30
hectares for crop X and 20 hectares for crop Y Example 8 (Manufacturing problem) A manufacturing company makes two models
A and B of a product |
1 | 6475-6478 | Fig 12 8
Hence, the society will get the maximum profit of Rs 4,95,000 by allocating 30
hectares for crop X and 20 hectares for crop Y Example 8 (Manufacturing problem) A manufacturing company makes two models
A and B of a product Each piece of Model A requires 9 labour hours for fabricating
and 1 labour hour for finishing |
1 | 6476-6479 | 8
Hence, the society will get the maximum profit of Rs 4,95,000 by allocating 30
hectares for crop X and 20 hectares for crop Y Example 8 (Manufacturing problem) A manufacturing company makes two models
A and B of a product Each piece of Model A requires 9 labour hours for fabricating
and 1 labour hour for finishing Each piece of Model B requires 12 labour hours for
fabricating and 3 labour hours for finishing |
1 | 6477-6480 | Example 8 (Manufacturing problem) A manufacturing company makes two models
A and B of a product Each piece of Model A requires 9 labour hours for fabricating
and 1 labour hour for finishing Each piece of Model B requires 12 labour hours for
fabricating and 3 labour hours for finishing For fabricating and finishing, the maximum
labour hours available are 180 and 30 respectively |
1 | 6478-6481 | Each piece of Model A requires 9 labour hours for fabricating
and 1 labour hour for finishing Each piece of Model B requires 12 labour hours for
fabricating and 3 labour hours for finishing For fabricating and finishing, the maximum
labour hours available are 180 and 30 respectively The company makes a profit of
Rs 8000 on each piece of model A and Rs 12000 on each piece of Model B |
1 | 6479-6482 | Each piece of Model B requires 12 labour hours for
fabricating and 3 labour hours for finishing For fabricating and finishing, the maximum
labour hours available are 180 and 30 respectively The company makes a profit of
Rs 8000 on each piece of model A and Rs 12000 on each piece of Model B How many
pieces of Model A and Model B should be manufactured per week to realise a maximum
profit |
1 | 6480-6483 | For fabricating and finishing, the maximum
labour hours available are 180 and 30 respectively The company makes a profit of
Rs 8000 on each piece of model A and Rs 12000 on each piece of Model B How many
pieces of Model A and Model B should be manufactured per week to realise a maximum
profit What is the maximum profit per week |
1 | 6481-6484 | The company makes a profit of
Rs 8000 on each piece of model A and Rs 12000 on each piece of Model B How many
pieces of Model A and Model B should be manufactured per week to realise a maximum
profit What is the maximum profit per week Corner Point Z = 10500x + 9000y
O(0, 0)
0
A( 40, 0)
420000
B(30, 20)
495000
C(0,50)
450000
← Maximum
© NCERT
not to be republished
518
MATHEMATICS
←
Solution Suppose x is the number of pieces of Model A and y is the number of pieces
of Model B |
1 | 6482-6485 | How many
pieces of Model A and Model B should be manufactured per week to realise a maximum
profit What is the maximum profit per week Corner Point Z = 10500x + 9000y
O(0, 0)
0
A( 40, 0)
420000
B(30, 20)
495000
C(0,50)
450000
← Maximum
© NCERT
not to be republished
518
MATHEMATICS
←
Solution Suppose x is the number of pieces of Model A and y is the number of pieces
of Model B Then
Total profit (in Rs) = 8000 x + 12000 y
Let
Z = 8000 x + 12000 y
We now have the following mathematical model for the given problem |
1 | 6483-6486 | What is the maximum profit per week Corner Point Z = 10500x + 9000y
O(0, 0)
0
A( 40, 0)
420000
B(30, 20)
495000
C(0,50)
450000
← Maximum
© NCERT
not to be republished
518
MATHEMATICS
←
Solution Suppose x is the number of pieces of Model A and y is the number of pieces
of Model B Then
Total profit (in Rs) = 8000 x + 12000 y
Let
Z = 8000 x + 12000 y
We now have the following mathematical model for the given problem Maximise Z = 8000 x + 12000 y |
1 | 6484-6487 | Corner Point Z = 10500x + 9000y
O(0, 0)
0
A( 40, 0)
420000
B(30, 20)
495000
C(0,50)
450000
← Maximum
© NCERT
not to be republished
518
MATHEMATICS
←
Solution Suppose x is the number of pieces of Model A and y is the number of pieces
of Model B Then
Total profit (in Rs) = 8000 x + 12000 y
Let
Z = 8000 x + 12000 y
We now have the following mathematical model for the given problem Maximise Z = 8000 x + 12000 y (1)
subject to the constraints:
9x + 12y ≤ 180 (Fabricating constraint)
i |
1 | 6485-6488 | Then
Total profit (in Rs) = 8000 x + 12000 y
Let
Z = 8000 x + 12000 y
We now have the following mathematical model for the given problem Maximise Z = 8000 x + 12000 y (1)
subject to the constraints:
9x + 12y ≤ 180 (Fabricating constraint)
i e |
1 | 6486-6489 | Maximise Z = 8000 x + 12000 y (1)
subject to the constraints:
9x + 12y ≤ 180 (Fabricating constraint)
i e 3x + 4y ≤ 60 |
1 | 6487-6490 | (1)
subject to the constraints:
9x + 12y ≤ 180 (Fabricating constraint)
i e 3x + 4y ≤ 60 (2)
x + 3y ≤ 30
(Finishing constraint) |
1 | 6488-6491 | e 3x + 4y ≤ 60 (2)
x + 3y ≤ 30
(Finishing constraint) (3)
x ≥ 0, y ≥ 0
(non-negative constraint) |
1 | 6489-6492 | 3x + 4y ≤ 60 (2)
x + 3y ≤ 30
(Finishing constraint) (3)
x ≥ 0, y ≥ 0
(non-negative constraint) (4)
The feasible region (shaded) OABC determined by the linear inequalities (2) to (4)
is shown in the Fig 12 |
1 | 6490-6493 | (2)
x + 3y ≤ 30
(Finishing constraint) (3)
x ≥ 0, y ≥ 0
(non-negative constraint) (4)
The feasible region (shaded) OABC determined by the linear inequalities (2) to (4)
is shown in the Fig 12 9 |
1 | 6491-6494 | (3)
x ≥ 0, y ≥ 0
(non-negative constraint) (4)
The feasible region (shaded) OABC determined by the linear inequalities (2) to (4)
is shown in the Fig 12 9 Note that the feasible region is bounded |
1 | 6492-6495 | (4)
The feasible region (shaded) OABC determined by the linear inequalities (2) to (4)
is shown in the Fig 12 9 Note that the feasible region is bounded Fig 12 |
1 | 6493-6496 | 9 Note that the feasible region is bounded Fig 12 9
Let us evaluate the objective function Z at each corner point as shown below:
Corner Point
Z = 8000 x + 12000 y
0 (0, 0)
0
A (20, 0)
160000
B (12, 6)
168000
Maximum
C (0, 10)
120000
We find that maximum value of Z is 1,68,000 at B (12, 6) |
1 | 6494-6497 | Note that the feasible region is bounded Fig 12 9
Let us evaluate the objective function Z at each corner point as shown below:
Corner Point
Z = 8000 x + 12000 y
0 (0, 0)
0
A (20, 0)
160000
B (12, 6)
168000
Maximum
C (0, 10)
120000
We find that maximum value of Z is 1,68,000 at B (12, 6) Hence, the company
should produce 12 pieces of Model A and 6 pieces of Model B to realise maximum
profit and maximum profit then will be Rs 1,68,000 |
1 | 6495-6498 | Fig 12 9
Let us evaluate the objective function Z at each corner point as shown below:
Corner Point
Z = 8000 x + 12000 y
0 (0, 0)
0
A (20, 0)
160000
B (12, 6)
168000
Maximum
C (0, 10)
120000
We find that maximum value of Z is 1,68,000 at B (12, 6) Hence, the company
should produce 12 pieces of Model A and 6 pieces of Model B to realise maximum
profit and maximum profit then will be Rs 1,68,000 © NCERT
not to be republished
LINEAR PROGRAMMING 519
EXERCISE 12 |
1 | 6496-6499 | 9
Let us evaluate the objective function Z at each corner point as shown below:
Corner Point
Z = 8000 x + 12000 y
0 (0, 0)
0
A (20, 0)
160000
B (12, 6)
168000
Maximum
C (0, 10)
120000
We find that maximum value of Z is 1,68,000 at B (12, 6) Hence, the company
should produce 12 pieces of Model A and 6 pieces of Model B to realise maximum
profit and maximum profit then will be Rs 1,68,000 © NCERT
not to be republished
LINEAR PROGRAMMING 519
EXERCISE 12 2
1 |
1 | 6497-6500 | Hence, the company
should produce 12 pieces of Model A and 6 pieces of Model B to realise maximum
profit and maximum profit then will be Rs 1,68,000 © NCERT
not to be republished
LINEAR PROGRAMMING 519
EXERCISE 12 2
1 Reshma wishes to mix two types of food P and Q in such a way that the vitamin
contents of the mixture contain at least 8 units of vitamin A and 11 units of
vitamin B |
1 | 6498-6501 | © NCERT
not to be republished
LINEAR PROGRAMMING 519
EXERCISE 12 2
1 Reshma wishes to mix two types of food P and Q in such a way that the vitamin
contents of the mixture contain at least 8 units of vitamin A and 11 units of
vitamin B Food P costs Rs 60/kg and Food Q costs Rs 80/kg |
1 | 6499-6502 | 2
1 Reshma wishes to mix two types of food P and Q in such a way that the vitamin
contents of the mixture contain at least 8 units of vitamin A and 11 units of
vitamin B Food P costs Rs 60/kg and Food Q costs Rs 80/kg Food P contains
3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains
4 units/kg of Vitamin A and 2 units/kg of vitamin B |
1 | 6500-6503 | Reshma wishes to mix two types of food P and Q in such a way that the vitamin
contents of the mixture contain at least 8 units of vitamin A and 11 units of
vitamin B Food P costs Rs 60/kg and Food Q costs Rs 80/kg Food P contains
3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains
4 units/kg of Vitamin A and 2 units/kg of vitamin B Determine the minimum cost
of the mixture |
1 | 6501-6504 | Food P costs Rs 60/kg and Food Q costs Rs 80/kg Food P contains
3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains
4 units/kg of Vitamin A and 2 units/kg of vitamin B Determine the minimum cost
of the mixture 2 |
1 | 6502-6505 | Food P contains
3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains
4 units/kg of Vitamin A and 2 units/kg of vitamin B Determine the minimum cost
of the mixture 2 One kind of cake requires 200g of flour and 25g of fat, and another kind of cake
requires 100g of flour and 50g of fat |
1 | 6503-6506 | Determine the minimum cost
of the mixture 2 One kind of cake requires 200g of flour and 25g of fat, and another kind of cake
requires 100g of flour and 50g of fat Find the maximum number of cakes which
can be made from 5kg of flour and 1 kg of fat assuming that there is no shortage
of the other ingredients used in making the cakes |
1 | 6504-6507 | 2 One kind of cake requires 200g of flour and 25g of fat, and another kind of cake
requires 100g of flour and 50g of fat Find the maximum number of cakes which
can be made from 5kg of flour and 1 kg of fat assuming that there is no shortage
of the other ingredients used in making the cakes 3 |
1 | 6505-6508 | One kind of cake requires 200g of flour and 25g of fat, and another kind of cake
requires 100g of flour and 50g of fat Find the maximum number of cakes which
can be made from 5kg of flour and 1 kg of fat assuming that there is no shortage
of the other ingredients used in making the cakes 3 A factory makes tennis rackets and cricket bats |
1 | 6506-6509 | Find the maximum number of cakes which
can be made from 5kg of flour and 1 kg of fat assuming that there is no shortage
of the other ingredients used in making the cakes 3 A factory makes tennis rackets and cricket bats A tennis racket takes 1 |
1 | 6507-6510 | 3 A factory makes tennis rackets and cricket bats A tennis racket takes 1 5 hours
of machine time and 3 hours of craftman’s time in its making while a cricket bat
takes 3 hour of machine time and 1 hour of craftman’s time |
1 | 6508-6511 | A factory makes tennis rackets and cricket bats A tennis racket takes 1 5 hours
of machine time and 3 hours of craftman’s time in its making while a cricket bat
takes 3 hour of machine time and 1 hour of craftman’s time In a day, the factory
has the availability of not more than 42 hours of machine time and 24 hours of
craftsman’s time |
1 | 6509-6512 | A tennis racket takes 1 5 hours
of machine time and 3 hours of craftman’s time in its making while a cricket bat
takes 3 hour of machine time and 1 hour of craftman’s time In a day, the factory
has the availability of not more than 42 hours of machine time and 24 hours of
craftsman’s time (i) What number of rackets and bats must be made if the factory is to work
at full capacity |
1 | 6510-6513 | 5 hours
of machine time and 3 hours of craftman’s time in its making while a cricket bat
takes 3 hour of machine time and 1 hour of craftman’s time In a day, the factory
has the availability of not more than 42 hours of machine time and 24 hours of
craftsman’s time (i) What number of rackets and bats must be made if the factory is to work
at full capacity (ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find
the maximum profit of the factory when it works at full capacity |
1 | 6511-6514 | In a day, the factory
has the availability of not more than 42 hours of machine time and 24 hours of
craftsman’s time (i) What number of rackets and bats must be made if the factory is to work
at full capacity (ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find
the maximum profit of the factory when it works at full capacity 4 |
1 | 6512-6515 | (i) What number of rackets and bats must be made if the factory is to work
at full capacity (ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find
the maximum profit of the factory when it works at full capacity 4 A manufacturer produces nuts and bolts |
1 | 6513-6516 | (ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find
the maximum profit of the factory when it works at full capacity 4 A manufacturer produces nuts and bolts It takes 1 hour of work on machine A
and 3 hours on machine B to produce a package of nuts |
1 | 6514-6517 | 4 A manufacturer produces nuts and bolts It takes 1 hour of work on machine A
and 3 hours on machine B to produce a package of nuts It takes 3 hours on
machine A and 1 hour on machine B to produce a package of bolts |
1 | 6515-6518 | A manufacturer produces nuts and bolts It takes 1 hour of work on machine A
and 3 hours on machine B to produce a package of nuts It takes 3 hours on
machine A and 1 hour on machine B to produce a package of bolts He earns a
profit of Rs17 |
1 | 6516-6519 | It takes 1 hour of work on machine A
and 3 hours on machine B to produce a package of nuts It takes 3 hours on
machine A and 1 hour on machine B to produce a package of bolts He earns a
profit of Rs17 50 per package on nuts and Rs 7 |
1 | 6517-6520 | It takes 3 hours on
machine A and 1 hour on machine B to produce a package of bolts He earns a
profit of Rs17 50 per package on nuts and Rs 7 00 per package on bolts |
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