The only grocery store in a small rural community carries two brands of frozen apple juice,a local brand that it obtains at the cost of 18 cents per can and a well-known national brand that it obtains at the cost of 40 cents per can.The grocer estimates that if the local brand is sold for x cents per can and the national brand for y cents per can,approximately 70 - 5x + 4y cans of the local brand and 80 + 6x - 7y cans of the national brand will be sold each day.How should the grocer price each brand to maximize the profit from the sale of the juice?
A)local brand (x)at 39 cents,national brand (y)at 104 cents
B)local brand (x)at 46 cents,national brand (y)at 52 cents
C)local brand (x)at 44 cents,national brand (y)at 52 cents
D)local brand (x)at 88 cents,national brand (y)at 50 cents

Question

Quizplus · Accepted Answer

The profit from selling one can of the local brand is (x-18) cents and for the national brand is (y-40) cents. The total profit from selling (70-5x+4y) cans of the local brand and (80+6x-7y) cans of the national brand is:

Profit = (70-5x+4y)(x-18) + (80+6x-7y)(y-40)

Simplifying this expression gives:

Profit = -5x^2 - 23xy + 118x - 28y^2 - 1840 + 240y

To maximize the profit, we need to differentiate the expression above for profit with respect to x and y, and we set the derivatives equal to zero.

dProfit/dx = -10x -23y + 118 = 0
dProfit/dy = -23x - 56y + 240 = 0

Solving these two equations simultaneously gives x = 44 and y = 52. These values of x and y correspond to choice C. Therefore, the grocer should price the local brand at 44 cents and the national brand at 52 cents to maximize the profit from the sale of the juice.

The Only Grocery Store in a Small Rural Community Carries