Question

Suppose the Sunglasses Hut Company has a profit function given by P(q)=−0.03q2+4q−50P(q)=-0.03q2+4q-50, where qq is the number of thousands of pairs of sunglasses sold and produced, and P(q)P(q) is the total profit, in thousands of dollars, from selling and producing qq pairs of sunglasses.

A) Find a simplified expression for the marginal profit function. (Be sure to use the proper variable in your answer.)

MP(q)=MP(q)=    

B) 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.

C) 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 1

Given, P(q)=−0.03q²+4q−50 , 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) Marginal profit ,MP(q) is given by the change in profit P(q) with unit change in quantity q

So MP(q) = d P(q) / dq = - 0.06 q + 4 [ Differentiating −0.03q²+4q−50 with respect to q ]

(B) To maximize profits, dP(q) / dq = 0 [ Profits increases with increase in q till a point of maxima is reached and then it declines. We need to find q at which this maxima is achieved]

So dP(q) / dq = - 0.06 q + 4 = 0

or q = 400/6 = 200/3 = 66.667 pairs of sunglasses (in thousands)

d²P(q) / dq² = - 0.06 < 0 => Profit is maximum for q = 66.667

So 66.667 pairs of sunglasses (in thousands) should be sold to maximize profits

(C) Putting q = 66.667 in P(q)=−0.03q²+4q−50, we get

P(q) = -0.03 * 66.667² + 4 * 66.667 - 50

= - 133.334 + 266.668 - 50

= 83.334

So, actual maximum profits (in thousands of dollars) that can be expected = 83.334