Question

Mr. Smith decides to feed his pet Doberman pinscher a combination of two dog foods.

Each can of brand A contains 4 units of​ protein, 1 unit of​ carbohydrates, and 2 units of fat and costs 70 cents. Each can of brand B contains 1 unit of​protein, 1 unit of​ carbohydrates, and 4 units of fat and costs 30 cents. Mr. Smith feels that each day his dog should have at least 9 units of​ protein, 6 units of​ carbohydrates, and 20 units of fat. How many cans of each dog food should he give to his dog each day to provide the minimum requirements at the least​ cost?

Question:

Mr.Smith should give his dog ________ ​can(s) of brand A and ___________ ​can(s) of brand B to provide the minimum requirements at the least cost.

Answer 1

Let x be the number of cans of Brand A and y be the number of cans of Brand B .

Objective function : C= 70x +30y = 0.70x + 0.30y

Inequalities of the given question is as follows:

For Protein : 4x +y 9

For Carbohydrates : x +y 6

For Fat : 2x +4y 20

x 0

y 0

Now , plot these equations :

For 4x+y 9 : When x=0 , y=9 And when y=0, x=2.25.

For x+y 6 : When x=0 , y=6 And when y=0 , x=6.

For 2x +4y 20 : When x=0 , y=5 And When y=0 , x=10.

By plotting these we get the following graph:

Now , we have 4 feasible points , A, B , C and D . No w, calculate the objective function value at each point:

A : (0,9) = 0.70(0) + (0.30)(9) = $ 2.7

B : (1,5) = (0.70)(1)+ (0.30)(5)= 0.70 +1.5 =$ 2.2

C : (2,4)= (0.70)(2)+ (0.30)(4)= 1.4 +1.2 =$ 2.6

D : (10,00= (0.70)(10)+ (0.30)(0)= $ 7

Because cost is minimizing at point B i.e (1,5) . This implies that Mr. Smith should give his dog 1 can of Brand A and 5 cans of Brand B to provide the minimum requirements at least cost.