Question

suppose the sunglasses hut company has a profit function given by , where is the number of thousands of pairs of sunglasses sold and produced, and is the total profit, in thousands of dollars, from selling and producing 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 1

The profit function for the Sunglasses Hut Company is given by
`[P(q) = \left( -0.03q^2 + 5q - 42 \right) \cdot 10^(-3),]` where q is the number of thousands of pairs of sunglasses sold and produced.

To find a simplified expression for the marginal profit function, we can take the derivative of P with respect to q.

The marginal profit function is given by
`[MP(q) = \left( -0.06q + 5 \right) \cdot 10^(-3).]`

To find the quantity of sunglasses that maximizes profits, we can set the marginal profit function to zero and solve for q.

We get
`[q = (5)/(0.06) = \boxed{83\text{ thousand pairs}}.]`

The maximum profit is then P=(−0.03(83) ^2+5(83)−42)⋅10−3= −43.6 thousand dollars .