Question

(From Textbook) A company makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat. If chicken costs 10 cents per ounce and grain costs 1 cent per ounce, how many ounces of each should the company use in each bag of dog food in order to minimize cost?

Answer 1

The company use 0 ounces of chicken and 100 ounces of grain in each bag of dog food in order to minimize cost.

Explanation:

Let x be the number of ounces of chicken

let y be the number of ounces of grains

We are given that Chicken has 10 grams of protein and 5 grams of fat per ounce.

Chicken has protein in 1 ounce = 10

Chicken has protein in x ounces = 10x

Chicken has fat in 1 ounce = 5

Chicken has fat in x ounces = 5x

We are also given that grain has 2 grams of protein and 2 grams of fat per ounce

Grain has protein in 1 ounce = 2

Grain has protein in y ounces = 2y

Grain has fat in 1 ounce = 2

Grain has fat in y ounces =2y

Now we are given that . A bag of dog food must contain at least 200 grams of protein and at least 150 grams of fat.

So, equation becomes:

`10x+2y\geq 200` ---A

`5x+2y\geq 150` ---B

Since x and y represent the ounces .

So,
`x\geq 0` ---C and
`y\geq 0` ----D

Plot A ,B ,C and D on the graph

Refer the attached graph so , the corner points are (0,100),(10,50) and (30,0)

We are also given that chicken costs 10 cents per ounce and grain costs 1 cent per ounce

So, Cost function becomes :
`c=10x+y`

At(0,100)

`c=10x+y`

`c=10(0)+100`

`c=100`

At(10,50)

`c=10x+y`

`c=10(10)+50`

`c=150`

At(30,0)

`c=10x+y`

`c=10(30)+0`

`c=300`

Minimum cost is 100 cents at (0,100)

So, the company use 0 ounces of chicken and 100 ounces of grain in each bag of dog food in order to minimize cost.