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 65% mild cheddar and 35% extra sharp, and the Zesty blend contains 75% mild cheddar and 25% extra sharp. This year, a local dairy cooperative offered to provide up to 8100 pounds of mild cheddar cheese for $1.30 per pound and up to 3500 pounds of extra sharp cheddar cheese for $1.50 per pound. The cost to blend and package the cheese spreads, excluding the cost of the cheese, is $0.30 per container. If each container of Regular is sold for $1.80 and each container of Zesty is sold for $2.10, how many containers of Regular and Zesty should New England Cheese produce

Answer 1

To determine the number of containers of Regular and Zesty cheese spreads that New England Cheese should produce, we need to set up a linear programming problem and solve it to find the optimal production quantities that maximize profit.

Step-by-step explanation:

To determine the number of containers of Regular and Zesty cheese spreads that New England Cheese should produce, we need to find the optimal production quantities that maximize profit. Let's start by setting up the problem.

Let x be the number of containers of Regular cheese spreads and y be the number of containers of Zesty cheese spreads.

The cost to blend and package each container is $0.30, so the total cost to blend and package the Regular cheese spreads is 0.30x and the total cost to blend and package the Zesty cheese spreads is 0.30y.

The revenue from selling each container of Regular is $1.80, so the total revenue from selling the Regular cheese spreads is 1.80x. Similarly, the revenue from selling each container of Zesty is $2.10, so the total revenue from selling the Zesty cheese spreads is 2.10y.

The cost of mild cheddar cheese is $1.30 per pound, and New England Cheese can use up to 8100 pounds of mild cheddar. The cost of extra sharp cheddar cheese is $1.50 per pound, and they can use up to 3500 pounds of extra sharp cheddar.

Let's set up the constraints for the amount of cheese used:

Mild cheddar cheese constraint: 0.65x + 0.75y ≤ 8100

Extra sharp cheddar cheese constraint: 0.35x + 0.25y ≤ 3500

Since we want to maximize profit, our objective function is the total profit:

Total profit = Total revenue - Total cost = (1.80x + 2.10y) - (0.30x + 0.30y) = 1.50x + 1.80y

Now, we can solve this linear programming problem using graphing, substitution, or matrix methods to find the optimal values of x and y that maximize profit.

Once we calculate the values of x and y, we can determine the number of containers of Regular and Zesty cheese spreads that New England Cheese should produce.

Answer 2

They would produce 14,400 packages of zetsy and the regular profit making would be $11,340.

Step-by-step explanation:

12-ounce

regular = 65%

mild = 35%

extra sharp zesty blend = 75%

extra sharp mild = 25%

8,100 mild for = $1.30

3,500 extra sharp = $1.50

cost blend and package = $0.30

regular sold = $1.8

zesty = $2.10

first we calculate the cost of each product

conversion Of ounces to pounds:

1 pound = 16 ounces

regular cost:

12/16 pounds x 65% mild x $1.30

= 0.63375

12/16 pounds x 35% sharp x $1.50

= 0.39375‬

+ blend and package $0.30

total = $1.3275‬

contribution: $1.8 – $1.3275

= $0,4725‬

zesty cost:

12/16 pounds x 75% mild x $1.30

= 0.73125‬

12/16 pounds x 25% sharp x $1.50

= 0.28125‬

+ blend and package 0.30

total $1,3125‬

contribution: $2.10 – $1.3125

= $0,7875‬

to maximize profit:

profit = $0.4725Regular + $0.7875zetzy

adding the constraint:

mild = Regular x $0.4775 + Zetzy x $0.5625 = 8,100

and

sharp = Regular x $0.2625 + Zetzy x $0.1875 = 3,100

contribution generated by each product/pound:

Regular Zesty

CM $ 0.47250 $ 0.47250

Constrain resource

( pounds of mild) 0.48750 0.56250

CM per constrain $ 0.9692 $ 1.4000

Constrain resource

( pounds of sharp) 0.26250 0.18750

CM per constrain $ 1.80 $ 4.20

As zesty is better we produce as much as we can:

8,100 pounts / 0.5625 per cheese spreads: 14,400

check for possibilities of other constraint:

14,400 zesty packages x 0.1875 pounds each = 2,700 pounds

because it’s possible we only produce zetsy

profit = $0.4725 (0) + $0.7875 (14,400)

profit = $ 11,340