A firm sells its product in a perfectly competitive market where other firms charge a price of $80 per unit

If the total costs are known, take the derivative to determine the marginal cost. Set the price equal to marginal cost and solve for quantity. Since the quantity can't be negative, find the average total cost which is the total cost divided by output, set the marginal cost equal to the average total cost and solve for quantity. In perfect competition the market price is the firm's price. The firm should produce 70/44 units of output. In a perfectly competitive market the firm charges the market price which is $80.

If the Total Costs C(Q) = 70 - 8Q - 2Q^2

Average Total Cost = 70/Q - 8 - 2Q

Marginal Cost = - 8 - 4Q

P = 80

P = MC

80 = -8 - 4Q

4Q = -88

2Q = -44

MC = ATC

-8 - 4Q = 70/Q - 8 - 2Q

-8 + 88 = 70/Q - 8 + 44

80 = 70/Q + 36

44 = 70/Q

Q = 70/44