college math problems

Question

Find the quantity a that maximizes profit if the total revenue, R(9), and the total cost, C(9), are given in dollars by

R(q) = 409q - 8q^2

C(q) = 1500 + 73q.

What is the maximum profit? Prove your answer.

Answer 1

The profit function P is given by:

P(q) = R(q) - C(q)

= 409q - 8q² - (1500 + 73q)

= 409q - 8q² - 1500 - 73q

= - 8q² + 336q - 1500

The derivative of the profit function is given by:

P'(q) = -8 * 2q + 336 = -16q + 336

P'(q) = 0

⇒ -16q + 336 = 0

⇒ -16q + 336 - 336 = 0 - 336

⇒ -16q = - 336

⇒ -16q/(-16) = - 336/(-16)

⇒ q = 21

The second derivative of the profit function is given by:

P''(q) = -16

Since the second derivative is negative everywhere (and in particular at q = 21),

at the point where q = 21, we have a local maximum.

Moreover, since the shape of the function P is a downward parabola,

at the point where q = 21 it reaches its global maximum.

So the quantity that maximizes profit is 21.

The value of the maximum profit is given by:

P(21) = - 8*21² + 336 * 21 - 1500 = 2,028