Assume that a competitive firm has the total cost function:

TC=1q^340q^2+710q+1700 (^ is to the power)

Suppose the price of the firm's output (sold in integer units) is $550 per unit.

Using calculus and formulas to find a solution (don't just build a table in a spreadsheet as in the previous lesson), what is the total profit at the optimal integer output level?

Please specify your answer as an integer.

Hint 1:The first derivative of the total cost function, which is cumulative, is the marginal cost function, which is incremental. The narrated lecture and formula summary explain how to compute the derivative.

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, where a, b, and c are constants.

Use the quadratic formula to solve for q:

q=(b b24ac)/2a (b2-4ac is under the square root sign )

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

Hint 2:When computing the total cost component of total profit for each candidate quantity, use the total cost function provided in the exercise statement (rather than summing the marginal costs using the marginal cost function).