Question

The New England Cheese Company produces two cheese spreads by blending mild cheddar cheese with extra sharp cheddar cheese. The cheese spreads are packaged in 12-ounce containers, which are then sold to distributors throughout the Northeast. The Regular blend contains 80% mild cheddar and 20% extra sharp, and the Zesty blend contains 60% mild cheddar and 40% extra sharp. This year, a local dairy cooperative offered to provide up to 8100 pounds of mild cheddar cheese for $1.20 per pound and up to 3000 pounds of extra sharp cheddar cheese for $1.40 per pound. The cost to blend and package the cheese spreads, excluding the cost of the cheese, is $0.20 per container. If each container of Regular is sold for $1.95 and each container of Zesty is sold for $2.20, how many containers of Regular and Zesty should New England Cheese produce?

Answer 1

Answer:

37 packs of Regular and 20 of Zesty

Step-by-step explanation:

Let x and y the number of Regular and Zesty packages that are to be produced.

Amount of mild cheddar in ounces needed for the production

80% of 12 ounces times x + 60% of 12 ounces times y

9.6x + 7.2y

Amount of extra sharp in ounces needed for the production

20% of 12 ounces times x + 40% of 12 ounces times y

2.4x + 4.8y

There are 16 ounces in a pound, so the total cost of blending is

(9.6x + 7.2y)(1.2/16) + (2.4x + 4.8y)(1.4/16)        

and packaging

0.2(x+y)  

Total cost (blending + packaging)

(9.6x + 7.2y)(1.2/16) + (2.4x + 4.8y)(1.4/16) + 0.2(x+y) =

1.13x + 1.16y

If we sold all the production, the profit (= Income - Cost of production) would be

1.95x + 2.2y -(1.13x+1.16y) = 0.82x + 1.04y

So we want to find the maximum of the linear function

P(x, y) = 0.82x + 1.04y

Now, let's find out the feasible region

“a local dairy cooperative offered to provide up to 8100 pounds of mild cheddar and up to 3000 pounds of extra sharp”

so (remember 16 ounces per pound)

16(9.6x + 7.2y) ≤ 8100

16(2.4x + 4.8y) ≤ 3000

or what is the same

9.6x + 7.2y ≤ 506.25

2.4x + 4.8y ≤ 187.5

We also have  

x ≥ 0

y ≥ 0

See picture for the feasible region

The vertex (37.25, 20.31) is the solution of the system

9.6x + 7.2y = 506.25

2.4x + 4.8y = 187.5

Since x, y must be integers, the maximum is attained in one of the points

(0, 39), (37, 20) or (52, 0)

Let's evaluate our function of profit on each of them

P(0,39) = 1.04*39 = 40.56

P(37,20) = 0.82*37+1.04*20 = 51.14

P(52,0) = 0.82*52 = 42.64

So the production that maximizes the profit is x=37, y=20