Problem 2-37 (Algorithmic) The New England Cheese Company produces two cheese sp

Question

Problem 2-37 (Algorithmic) 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 8,100 pounds of mild cheddar cheese for $1.20 per pound and up to 3,000 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? Do not round intermediate calculations. If required, round your answers to the nearest whole number. Let R number of containers of Regular Z = number of containers of Zesty Optimal Solution: R- profit = $

Explanation / Answer

Cost of mild cheddar cheese = 1.2/16 = $0.075 per ounce

Cost of extra sharp cheddar cheese = 1.4/16 = $0.0875 per ounce

12 ounce containers :

Regular -> 9.6 ounces = mild & 2.4 ounces = extra sharp

Zesty -> 7.2 ounces = mild and 4.8 ounces = extra sharp

Cost of making a 12 ounce Regular container = 9.6 x 0.075 + 2.4 x 0.0875 = 0.72 + 0.21 = $0.93

Cost of making a 12 ounce Zesty container = 7.2 x 0.075 + 4.8 x 0.0875 = 0.54 + 0.42 = $0.96

Cost per container = $0.2

We have to decide R and Z.

Profit on R regular containers = 1.95R - 0.2R - (0.93R) = 0.82R

Profit on Z zesty containers = 2.2Z - 0.2Z - (0.96Z) = 1.04Z

Profit = 0.82R + 1.04Z

Say, we use r pounds of 8100 pounds of Mild cheese for Regular.

That means, 8100 - r pounds of Mild cheese for Zesty.

Say, we use z pounds of 3000 pounds of extra sharp cheese for Regular.

That means, 3000 - z pounds of extra sharp cheese for Zesty.

FOR REGULAR :

r pounds = 16r ounces of Mild for R

For a container for R, we need 9.6 ounces of Mild.

No. of Regular containers = R = 16r/9.6

z pounds = 16z ounces of extra sharp for R

For a container for R, we need 2.4 ounces of extra sharp.

No. of Regular containers = R = 16z/2.4  

FOR ZESTY :

8100 - r pounds = 16*(8100-r) ounces of Mild for Z

For a container for Z, we need 7.2 ounces of Mild.

No. of Zesty containers = Z = 16*(8100-r)/7.2

3000 - z pounds = 16(3000-z) ounces of extra sharp for R

For a container for R, we need 4.8 ounces of extra sharp.

No. of Zesty containers = Z = 16(3000-z)/4.8  

We now have to equate the number of Z and R containers --->

16r/9.6 = 16z/2.4 -> r = 4z

16(8100-r)/7.2 = 16(3000-z)/4.8 -> 8100-r = (3000-z)*1.5

8100 - r = 4500 - 1.5z

Sub. r = 4z --->

8100 - 4z = 4500 - 1.5z

8100 - 4500 = 2.5z

3600 = 2.5z

z = 3600/2.5 = 1440 pounds

r = 4z = 4*1440 = 5760 pounds

ANSWER :

Therefore, the number of containers R = 16r/9.6 = 16*5760/9.6 = 9600

Therefore, the number of containers Z = 16(8100-r)/7.2 = 16*2340/7.2 = 5200

Profit = 0.82R + 1.04Z = 0.82(9600) + 1.04(5200) = 7872 + 5408 = $13280

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