Question

BeachWear Inc. manufactures beach sandals. Their expense and revenue functions are E=-300p + 32,000 and R= -275p^2 + 6,500p . Determine the maximum profit

to the nearest cent.

a. $10,036.36 b. $6764.45 c. $6409.09 d. $2945.45

Answer 1

Given,

Expense function:

$E=−300p+32,000$

Revenue function:

$R=−275p^2+6500p\\ Profit(π)=R−E\\π=−275p^2+6,500p−(−300p+32,000)\\π=−275p^2+6,500p+300p−32,000\\π=−275p^2+6,800p−32,000$

To maximize the profit function let us derivate the function with

 respect to p

$\frac{dπ}{dp}=−2×275p+6,800\\Set,\\ \frac{dπ}{dp}=0\\−2×275p+6,800=0\\6800=2×275p\\p=\frac{6800}{550}\\p=12.3636$

To check whether the p=12.3636 will give maximum profit or not let us find

 the second−order derivative

$\frac{d^2π}{dπ^2}=−2×275\\\frac{d^2π}{dπ^2}=−550<0$

Since the value of second−order derivative is less than zero p=12.3636 will

 give maximum profit

At p=12.3636 the profit will be maximum.

At p=12.3636 the value of revenue will be a sgiven below

$R=−275×(12.3636)^2+6,500×12.3636\\R=−42,036.1164+8036.34\\R=38,327.2836$

At p=12.3636 the value of expense will be a sgiven below

$E=−300××12.3636+32,000\\E=−3,709.08+32,000\\E=\$28,290.92\\So,\\Profit=R−E\\Profit=38,327.2836−28,290.92\\Profit=\$10,036.36$

The correct option is A.