# Total cost function.

Assume that a competitive firm has the total cost function:

TC = 1q3 - 40q2 + 810q + 1500

Suppose the price of the firm's output (sold in integer units) is $700 per unit.

Using calculus and formulas (but no tables and restricting your use of spreadsheets to implementing the quadratic formula) to find a solution, what is the total profit at the optimal output level?

Please specify your answer as an integer.

Hint: The first derivative of the total cost function is the marginal cost function.

Set the marginal cost equal to the marginal revenue (price in this case) to define an equation for the optimal quantity q.

Rearrange the equation to the quadratic form aq2 + bq + c = 0.

Use the quadratic formula to solve for q:

For non-integer quantity, round up and down to find the optimal value.

© BrainMass Inc. brainmass.com October 10, 2019, 8:07 am ad1c9bdddfhttps://brainmass.com/economics/output-and-costs/competitive-firms-total-cost-function-608718

#### Solution Preview

Assume that a competitive firm has the total cost function:

TC = 1q^3 - 40q^2 + 810q + 1500

Taking the first derivative gives us:

MC = 3(q)^2 - 80 (q) + 810

Setting equal to MR gives us:

700 = 3(q)^2 - 80 (q) + 810

We want to set this ...

#### Solution Summary

The expert examines finding the optimal output to maximize profit given a firm's production function.