Question

A company that manufactures machine parts determines that q units of a part will be sold when the price is p = 110 – q dollars per unit. The total cost to produce those q units is C(g) dollars, where C(q) = q³ - 25q² + 2q +3,000

a. For what value of q is the profit maximized?
b. Find the consumer surplus at the level of production that corresponds to maximum profit.

Answer 1

To find the value of q that maximizes profit, we need to calculate the revenue and cost functions.

Revenue function:
R(q) = p*q = (110 - q)*q = 110q - q²

Cost function:
C(q) = q³ - 25q² + 2q + 3,000

Profit function:
P(q) = R(q) - C(q) = (110q - q²) - (q³ - 25q² + 2q + 3,000) = -q³ + 86q² + 108q - 3,000

To find the maximum profit, we need to take the derivative of the profit function and set it equal to zero:

P'(q) = -3q² + 172q + 108 = 0
Solving for q, we get q = 6 or q = 28.

To determine which value of q gives us the maximum profit, we need to take the second derivative of the profit function:

P''(q) = -6q + 172

When q = 6, P''(q) = 136 > 0, so the profit is