Average fixed cost at profit-maximizing output

Question

Please be as explicit as possible and break-down each step.

Problem 1.  Consider a firm operating in a perfectly competitive markt. Let the market determined price of output be $10 per unit.  Assume that the firm's short-run total cost and marginal cost surves have been estimated to be:

STC = 0.5q³ - 3q² + 10q + 6
                              
SMC = 1.5q² - 6q + 10

a. How much output will this firm produce ?

b. How much profit (or loss) is this firm making in the short run ?

c. What is the value of average fixed cost at this profit-maximizing output?

d. At what output will average variable cost be minimized?

e. What is the value of average variable cost at its minimum point?

f. If price falls to $5, how much out put will this firm produce?

Problem 2.  A firm's short-run margianl cost curve SMC = q² + 20q + 100.  Calculate the firm's short-run supply curve with q as a function of p, the market price.

Problem 3. A competitive firm has the following long-run total cost and marginal cost curves

LTC = q³ - 40q² + 430q

LMC = 3q² - 80q + 430

a. How much output will this firm produce?

b. What will the firm's price be at this output?

c. What will the firm's revenue be at this output?

d. Is the firm's price equal to its long-run average cost at this output?