Find the number of waterfalls that will maximize the manufactuer's proft?

Question

A manufacturer of pond waterfalls knows that the total cost C dollars of making X waterfalls is given by C=100x+0.1x^2+1200 and that the corresponding sales revenue R dollars is given by R=750x. Find the number of waterfalls that will maximize the manufacturer's profit. (hint: profit=revenue-cost). Give the profit as well.

Answer 1

Profit = Revenue - Cost

P(x) = 750x - (100x +0.1x² +1200)

P(x) = 750x - 100x - 0.1x² - 1200

P(x) = -0.1x² + 650x - 1200

A quadratic has a maximum at the vertex which is given from x = -b/2a = -650/(2)(-.1) = 3250

Number of waterfalls = 3250

P(3250) = -0.1(3250)² + 650(3250) - 1200 = 1055050 so profit = $1,055,050