76.9k views
2 votes
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.

1 Answer

3 votes

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.

User Grmartin
by
7.7k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories