Expert: Ahmed Salami - 9/22/2011

Question
Q1:Suppose the Sunglasses Hut Company has a profit function given by P(q)=-0.03q^2+3q-33, where q is the number of thousands of pairs of sunglasses sold and produced, and P(q) is the total profit, in thousands of dollars, from selling and producing q  pairs of sunglasses.

A) How many pairs of sunglasses (in thousands) should be sold to maximize profits? (If necessary, round your answer to three decimal places.)

Answer:[       ]thousand pairs of sunglasses need to be sold.

B) What are the actual maximum profits (in thousands) that can be expected? (If necessary, round your answer to three decimal places.)

Answer:[        ]thousand dollars of maximum profits can be expected

Answer
Hi Desmond,
For the profit function P = -0.03q² + 3q - 33, the marginal profit function;
dP/dq = -0.06q + 3
The profit is maximum when dP/dq = 0
So,
A) -0.06q + 3 = 0
0.06q = 3
q = 3/0.06 = 50

B) The maximum expected profit is therefore;
P = -0.03(50)² + 3(50) - 33
= -0.03(2500) + 150 - 33
= -75 + 150 - 33
= 42

Regards