You sell deluxe solar-powered flashlights, with a sliding price scale given by t

Question

You sell deluxe solar-powered flashlights, with a sliding price scale given by the function p(q) = 72 - 3q, where q is in units of flashlights and p(q) is in dollars. Your fixed costs are $90, and each flashlight has a marginal cost of $4. (i) Find a formula for the total revenue from selling q flashlights. (ii) What is your largest possible revenue? (iii) Find the quantity that maximizes profit, by comparing marginal revenue and marginal cost. (iv) Find a formula for the profit P(q) made in producing and selling q flashlights. (v) Using this profit function, find the quantity that maximizes profit. (Your answer should agree with part (c).)

Explanation / Answer

(i) R(q) = p * q = (72 – 3q)q = 72q – 3q^2

(ii) R’(q) = 72 – 6q = 0 means q = 12

Largest revenue = 72(12) – 3(12^2) = 432

(iii) MR(q) = 72 – 6q and MC = 4

For maximum profit, MR = MC

72 – 6q = 4

6q = 68

q = 68/6 = 34/3

(iv) C(q) = MC(q) dq = 4 dq = 4q

Pr(q) = R(q) – C(q) = 72q – 3q^2 – (90 + 4q) = 68q – 3q^2 – 90

(v) For maximum profit, Pr’(q) = 0

68 – 6q = 0 means 6q = 68 or q = 34/3

This agrees with the answer obtained in part (c).

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