Linear Programming

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New answer posted•a year ago

Linear Programming Class 12th

24. A dietician wishes to mix together two kinds of food, X and Y, in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contents of one kg of food are given below:

Food

Vitamin A

Vitamin B

Vitamin C

X

1

2

3

Y

2

2

1

One kg of food X costs ?. 16, and one kg of food Y costs ?. 20. Find the least cost of the mixture which will produce the required diet.

19 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

Let the mixture contain x kg of food X and y kg of food Y, respectively.

The mathematical formulation of the given problem can be written as given below:

$M i n i m i s e z = 16 x + 20 y \dots \dots . . (i)$

Subject to the constraints,

$\begin{array}{l} \end{array} x + 2 y \geq 10 \dots \dots \dots \dots . (i i)$

$x + y \geq 6 \dots \dots \dots \dots (i i i)$

$3 x + y \geq 8 \dots \dots \dots \dots . (i v)$

$x, y \geq 0 \dots \dots \dots \dots (v)$

The feasible region determined by the system of constraints is given below:

A (10, 0), B (2, 4), C (1, 5) and D (0, 8) are

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New answer posted•a year ago

Linear Programming Class 12th

23. A farmer mixes two brands, P and Q of cattle feed. Brand P, costing ?. 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing ?. 200 per bag, contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements for nutrients A, B and C are 18 units, 45 units and 24 units, respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag. What is the minimum cost of the mixture per bag?

6 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

Let the farmer mix x bags of brand P and y bags of brand Q, respectively

The given information can be compiled in a table as given below:

	Vitamin A (units/kg)	Vitamin B (units/kg)	Vitamin C (units/kg)	Cost (Rs/kg)
Food P	3	2.5	2	250
Food Q	1.5	11.25	3	200
Requirement (units/kg)	18	45	24

The given problem can be formulated as given below:

$\begin{array}{l} \end{array} M i n i m i s e z = 250 x + 200 y \dots \dots \dots \dots (i)$

$3 x + 1.5 y \geq 18 \dots \dots \dots \dots . . (i i)$

$2.5 x + 11.25 y \geq 45 \dots \dots \dots . . (i i i)$

$2 x + 3 y \geq 24 \dots \dots \dots \dots . . (i v)$

$x, y \geq 0 \dots \dots \dots . (v)$

The feasible region determined by the system of constraints is given below:

A (18, 0), B (9, 2), C (3, 6) and D (0

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New answer posted•a year ago

Linear Programming Class 12th

22. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?

7 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

Let the diet contain x and y packets of foods P and Q, respectively. Hence,

x ≥ 0 and y ≥ 0

The mathematical formulation of the given problem is given below:

$M a x i m i s e z = 6 x + 3 y \dots \dots \dots \dots . . (i)$

Subject to the constraints,

$\begin{array}{l} \end{array} 4 x + y \geq 80 \dots \dots \dots \dots . (i i)$

$x + 5 y \geq 115 \dots \dots \dots . (i i i)$

$3 x + 2 y \leq 150 \dots \dots \dots \dots (i v)$

$x, y \geq 0 \dots \dots \dots \dots \dots (v)$

The feasible region determined by the system of constraints is given below:

A (15, 20), B (40, 15) and C (2, 72) a

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New answer posted•a year ago

Linear Programming Class 12th

21. The corner points of the feasible region determined by the following system of linear inequalities:

$2 x + y \leq 10, x + 3 y \leq 15, x, y \geq 0$ are (0, 0), (5, 0), (3, 4) and (0, 5). Let $Z = p x + q y$ , 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

15 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

The maximum value of Z is unique.

It is given that the maximum value of Z occurs at two points, (3, 4) and (0, 5).

∴ Value of Z at (3, 4) = Value of Z at (0, 5)

$\begin{array}{l} \Rightarrow p (3) + q (4) = p (0) + q (5) \end{array}$

$\begin{array}{l} \Rightarrow 3 p + 4 q = 5 q \end{array}$

$\begin{array}{l} \Rightarrow q = 3 p \end{array}$

Hence, the correct answer is D.

Hence option (D) is correct.

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New answer posted•a year ago

Linear Programming Class 12th

20. There are two types of fertilizers 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 at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs ?. 6/kg and F2 costs ?. 5/kg, determine how much of each type of fertilizer should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

11 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

Let the farmer buy x kg of fertilizer F₁ and y kg of fertilizer F₂. Therefore,

x ≥ 0 and y ≥ 0

The given information can be complied in a table as follows.

	Nitrogen (%)	Phosphoric Acid (%)	Cost (Rs/kg)
F₁ (x)	10	6	6

F₂ (y)	5	10	5
Requirement (kg)	14	14

F₁ consists of 10% nitrogen and F₂ consists of 5% nitrogen. However, the farmer requires at least 14 kg of

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New answer posted•a year ago

Linear Programming Class 12th

19. 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 ?. 4 per unit food and F2 costs ?. 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. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

6 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

Let the diet contain x units of food F₁ and y units of food F₂. Therefore,

x ≥ 0 and y ≥ 0

The given information can be complied in a table as follows.

	Vitamin A (units)	Mineral (units)	Cost per unit (Rs)
Food F₁ (x)	3	4	4
Food F₂ (y)	6	3	6
Requirement	80	100

The cost of food F₁ is Rs 4 per unit and of Food F₂ is ? 6 per unit. Therefore, the constraints are

$\begin{array}{l} \end{array} 3 x + 6 y \geq 80$

$4 x + 3 y \geq 100$

$x, y \geq 0$

$T o t a l c o s t o f t h e d i e t, Z = 4 x + 6 y$

The mathema

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New answer posted•a year ago

Linear Programming Class 12th

18. 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 ? 70 lakhs and if his profit on the desktop model is ?. 4500 and on portable model is ?. 5000.

6 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

Let the merchant stock x desktop models and y portable models. Therefore,

x ≥ 0 and y ≥ 0

The cost of a desktop model is Rs 25000 and of a portable model is Rs 4000. However, the merchant can invest a maximum of Rs 70 lakhs.

$\begin{array}{l} ∴ 25000 x + 40000 y \leq 7000000 \end{array}$

$\begin{array}{l} 5 x + 8 y \leq 1400 \end{array}$

The monthly demand of computers will not ex

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New answer posted•a year ago

Linear Programming Class 12th

17. A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A requires 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 ?.5 each for type A and ?.6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?

7 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

Let the company manufacture x souvenirs of type A and y souvenirs of type B. Therefore,

x ≥ 0 and y ≥ 0

The given information can be complied in a table as follows.

	Type A	Type B	Availability
Cutting (min)	5	8	3 × 60 + 20 =200
Assembling (min)	10	8	4 × 60 = 240

The profit on type A souvenirs is Rs 5 and on type B souvenirs is Rs 6. Therefore, the constraints are

$\begin{array}{l} 5 x + 8 y \leq 200 \end{array}$

$\begin{array}{l} 10 x + 8 y \leq 240 i . e ., 5 x + 4 y \leq 120 \end{array}$

$\begin{array}{l} T o t a l p r o f i t, Z = 5 x + 6 y \end{array}$

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New answer posted•a year ago

Linear Programming Class 12th

16. 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. The profit from the sale of a lamp is ?.5 and that from a shade is ?.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 maximize his profit?

9 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

Let the cottage industry manufacture x pedestal lamps and y wooden shades. Therefore,

x ≥ 0 and y ≥ 0

The given information can be compiled in a table as follows.

	Lamps	Shades	Availability
Grinding/Cutting Machine (h)	2	1	12
Sprayer (h)	3	2	20

The profit on a lamp is Rs 5 and on the shades is Rs 3. Therefore, the constraints are

$\begin{array}{l} 2 x + y \leq 12 \end{array}$

$\begin{array}{l} 3 x + 2 y \leq 20 \end{array}$

$\begin{array}{l} T o t a l p r o f i t, Z = 5 x + 3 y \end{array}$

The mathematical fo

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New answer posted•a year ago

Linear Programming Class 12th

15. A factory manufacturers 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 screw A, while it takes 6 minutes on automatic and 3 minutes on 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 ?. 7 and screws B at a profit of ?. 10. Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce a day in order to maximize his profit? Determine the maximum profit.

5 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

Let the factory manufacture x screws of type A and y screws of type B on each day. Therefore,

x ≥ 0 and y ≥ 0

The given information can be compiled in a table as follows.

	Screw A	Screw B	Availability
Automatic Machine (min)	4	6	4 × 60 =240
Hand Operated Machine (min)	6	3	4 × 60 =240

The profit on a package of screws A is Rs 7 and on the package of screws B is Rs 10. Therefore, t

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Food	Vitamin A	Vitamin B	Vitamin C
X	1	2	3
Y	2	2	1

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