Question

An animal feed to be mixed from soybean meal and oats must contain at least 168 lb of protein, 27 lb of fat, and 14 lb of mineral ash. Each sack of soybeans costs $21 and contains 70 lb of protein, 9 lb of fat, and 7 lb of mineral ash. Each sack of oats costs $7 and contains 21 lb of protein, 7 lb of fat, and 1 lb of mineral ash. How many sacks of each should be used to satisfy the minimum requirements at minimum costs?What is minimum cost?How many sacks of each feed should be used to satisfy the minimum requirements at minimum cost?[____] sacks of soybean meal[____] sacks of oats

Answer 1

EXPLANATION

Let's see the facts:

Mandatory requirements:

168 lbs protein

27 lb fat

14 lb mineral ash

Soybeans cost = $21 ----> Contents:

70 lb ----> protein

9lb -------> fat

7 lb -------> mineral ash

Oats cost = $7-------------> Contents:

21 lb ----> protein

7 lb ------> fat

1 lb -------> mineral ash

Let's call x to the sacks of soybean and y to the sacks of oats, we have the following limitations:

(1) 70x + 21y >=168

(2) 9x + 7y >=27

(3) 7x + 1y >= 14

(4) x>=0

(5) y>= 0

The cost function is:

f(x,y) = 21x + 7y

This function has a minimum in one of the corners of the graph :

Isolating y in (1):

y >= 168/21 -(70x/21)

Isolating y in (2):

y >= 27/7 -9x/7

Isolating y in (3):

y >= 14 -7x

The corner points are: (3,0), (0,14) , (18/11,28/11), (29/6,)

F(3,0) = 21(3) + 7*0 = 63

F(0,14) = 21(0) + 7*14 = 98

F(18/11),28/11) = 21(18/11) + 7(28/11) = 52

F(29/6,-33/14) = 21(29/6)+7(-33/14) = 85

The minimum cost is achieved for 18/11 = 1.63 sacks of soybean meal and 28/11=2.54 sacks of oats.

The minimum cost is $52