Show and explain all calculations. Provide detailed economic reasoning.

1. A competitive firm's short run total cost function is given by

    TC = q³ - 16q² + 100q + 120

(a) Calculate the profit maximizing output and the resulting profit when price is $30.   

(b) Calculate the profit maximizing output and the resulting profit when price is $80.

2. In a perfectly competitive industry each producer has a long run total cost function given by

LTC = q³ - 24q² + 200q

where q denotes the output of the individual firm.

The market demand for the product is

Q = 1480 - 5P

where Q and P denote the market output and price respectively.

(a) Calculate the optimal output produced by each firm at the long run competitive equilibrium (LRCE).

(b) Calculate the market price and market output at the LRCE.

(c) Calculate the number of firms at the LRCE.