a. If the total cost is given, take the derivative of the total cost function to determine the marginal cost function. In perfect competition the equilibrium quantity is where the price is equal to marginal cost. The price is equal to the market price of $60. Set the price equal to the marginal cost and solve for Q. The output that the firm maximizes profit is 5.

In perfect competition set P = MC

TC = 100 + 10Q + 5Q^2

MC = 10 + 10Q

P = 60

60 = 10 + 10Q

10Q = 50

Q = 5

b. The total revenue is equal to the market price multiplied by the equilibrium quantity. To find the total cost, input the equilibrium quantity into the total cost function and solve for the total cost. The profit is equal to the total revenue minus the total cost. The total revenue is equal to $300, the total cost is equal to $275 and the profit is equal to $300 - $275 = $25.

Total Revenue = P * Q

TR = $60 * 5 = $300

TC = 100 + 10(5) + 5(5^2) = $100 + $50 + $125 = $275

Profit = $300 - $275 = $25

c. The average total cost is equal to the total cost divided by the equilibrium output. The average total cost is equal to $275/5 = $55.

ATC = $275/5 = $55

d. In perfect competition firms enter until there is zero economic profit. Since they have the same cost function and the price is the same entry will always be profitable. Therefore the market is in short run equilibrium but it is not in long run equilibrium until there is zero economic profit.