How many pairs of sunglasses in thousands should be sold to

Question

1. Given the given cost function

C(x)=8700 + 440x + 0.5x² and the demand function p(x)=1320 .

Find the production level that will maximaze profit.

2. A box with a square base and open top must have a volume of 186624 cm³ . We wish to find the dimensions of the box that minimize the amount of material used.

First, find a formula for the surface area of the box in terms of only x , the length of one side of the square base.

[Hint: use the volume formula to express the height of the box in terms of x .]

Simplify your formula as much as possible.

3. For the given cost function

C(x)=48400 + 600x + x²

find:

a) The production level that will minimize the average cost

b) The minimal average cost

4. A company's revenue from selling x units of an item is given as R = 1000x - 2x² . If sales are increasing at the rate of 55 units per day, how rapidly is revenue increasing (in dollars per day) when 210 units have been sold?

5. Suppose the Sunglasses Hut Company has a profit function given by P(q)= -0.03q² +5q^-38 , 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) Find a simplified expression for the marginal profit function. (Be sure to use the proper variable in your answer.)

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.)

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