Question

The profit p(x) of a cosmetics company in thousands of dollars is given by P(x) = -5x^2 + 400x - 2550, where x is the amount spent on advertising, in thousands of dollars. What amount must be spent on advertising to obtain a profit of at least $4 000 000?

Answer 1

-5x^2 + 400x - 2550 > 4000
5x^2 - 400x + 6550 < 0
x^2 - 80x + 1310 < 0

what are the x-intercepts?
x^2 - 80x + 1310 = 0
x^2 - 80x + 1600 = -1310 + 1600
(x-40)^2 = 290
x-40 = ± √290
x = 40±√290
= appr 57.029 or 22.97

They must spend between $22970 and $57029 to make a profit of 4,000,000

test:
let adv be = 22, 000
P(22) = 3830 < 4000 , not enough
let adv be 25,000
P(25) = 4325 > 4000 , that's good
let adv be 57000
P(57) = 4004 > 4000 , that's still good
let adv be 58000
P(58) = 3830 < 4000 , no good

Wolfram also confirms by conclusion
http://www.wolframalpha.com/input/?i=-5x%5E2+%2B+400x+-+2550+%3E+4000

Answer 2

help