Assume that a monopolist faces a demand curve for its product given by:

p=1102q

Further assume that the firm's cost function is:

TC=420+12q

Use calculus and formulas to find a solution (don't just build a table in a spreadsheet as in the previous lesson).

Round the optimal quantity to the nearest hundredth before computing the optimal price, which you should also round to the nearest hundredth. Use these rounded values to compute optimal profit. Note: Non-integer quantities may make sense when each unit of q represents a bundle of many individual items.

Hint 1:Define a formula for Total Revenue using the demand curve equation.

Hint 2:The first derivative of the total profit function, which is cumulative, is the marginal profit function, which is incremental. The lecture and formula summary explain how to compute the derivative.

Set the marginal profit equal to zero to define an equation for the optimal quantity q.

Hint 3:When computing the total profit for a candidate quantity, use the total profit function you define (rather than summing the marginal profits using the marginal profit function).

How much output should the firm produce?

What price should the monopolist choose to maximize profits?

What is the profit for the firm at the optimal quantity and price?