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

$p=70-3q$

Further assume that the firm's cost function is:

$TC=510+8q$

Use calculus and formulas to find a solution $($don$\text{'}$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?

Please round your answer to the nearest hundredth.

What price should the monopolist choose to maximize profits?

Follow the rounding guidance in the exercise statement for the optimal quantity when you compute the optimal

price. Please round your optimal price answer to the nearest hundredth.

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

Follow the rounding guidance in the exercise statement for quantity and price when you compute the optimal profit.

Please round your optimal profit answer to the nearest integer.