Explanation:
a. To find the value of q that maximizes profit, we need to find the value of q that maximizes revenue and then subtract the cost to produce that quantity. Revenue is equal to the price per unit multiplied by the quantity sold, or R(q) = q(110 - q). Thus, profit is given by P(q) = R(q) - C(q) = q(110 - q) - (q³ - 25q² + 2q + 3,000). Simplifying this expression, we get P(q) = -q³ + 85q² - 108q - 3,000. To find the value of q that maximizes profit, we need to take the derivative of P(q) with respect to q and set it equal to zero. This gives us -3q² + 170q - 108 = 0. Solving for q using the quadratic formula, we get q = 4.89 or q = 23.44. Since the company cannot produce a fractional quantity of parts, the value of q that maximizes profit is 23.
b. Consumer surplus is the difference between the maximum price that a consumer is willing to pay for a good and the actual price that they pay. In this case, the maximum price that a consumer is willing to pay is given by the demand function, p = 110 - q. At the level of production that corresponds to maximum profit, q = 23, so the price per unit is p = 110 - 23 = 87 dollars. The consumer surplus is then given by the integral of the demand function from q = 0 to q = 23, or ∫[0,23] (110 - q) dq = 23(110) - (23²/2) = 1,955 dollars.