The function P(x) = -30x^2 + 360x + 785 models the profit, P(x), earned by a theatre owner on the basis of a ticket price, x. Both the profit and the ticket price are in dollars. What is the maximum profit, and how much should the tickets cost?
Take the derivative of the profit function:
P'(x) = -60x + 360
We find the maximum when we set the profit derivative equal to 0
-60x + 360 = 0
Subtract 360 from both sides:
-60x = -360
Divide each side by -60
x = 6 <-- This is the ticket price to maximize profit
Substitute x = 6 into the profit equation:
P(6) = -30(6)^2 + 360(6) + 785
P(6) = -1080 + 2160 + 785
P(6) = 1865
Take the derivative of the profit function:
P'(x) = -60x + 360
We find the maximum when we set the profit derivative equal to 0
-60x + 360 = 0
Subtract 360 from both sides:
-60x = -360
Divide each side by -60
x = 6 <-- This is the ticket price to maximize profit
Substitute x = 6 into the profit equation:
P(6) = -30(6)^2 + 360(6) + 785
P(6) = -1080 + 2160 + 785
P(6) = 1865