Maximize profits

Question

Suppose the Sunglasses Hut Company's profit is given by the equation P=-0.02q²+3q-32, where P is the total profit, in thousands of dollars, and q is the total number of sunglasses sold, in thousands. For example, when q=10, the equation says that P=-4. This would mean that when 10000 sunglasses are sold, the profit is $-4000.

 

A) How many pairs of sunglasses (in thousands) should be sold to maximize profits?

Answer: ?
thousand pairs of sunglasses need to be sold.

 

B) What are the actual maximum profits (in thousands) that can be expected?

Answer: ?
thousand dollars of maximum profits can be expected.

Answer 1

P = -0.02q² + 3q - 32

 

To find the maximum profit, take the derivative of P wrt q  (dP/dq), set it to zero, then solve for q.  The value of q that you get is the number of sunglasses you need to sell (in thousands) to maximize profits.  To find the value of the maximum profit itself, plug the value of q you got into the profit equation and compute the max profit (remember, it's in thousands of dollars).