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

a) Determine the maximum profit the company can make
b) Determine the amount spent on advertising that will result in the maximum proift

Answer 1

they are asking for the second and first coordinate of the vertex respectively

Answer 2

the first coordinate of \[y=ax^2+bx+c\] is \[-\frac{b}{2a}\] and the second coordinate is what you get when you replace x by the first coordinate

Answer 3

@satellite73 to find the maximum profit, don't i have to use the quadratic formula?

Answer 4

no

Answer 5

wait hold on a second

Answer 6

there is a minus sign in front right?\[ P(x)=-5x^2+400x-2550\]

Answer 7

other wise it would not have a maximum, it would have a minimum
to set it equal to zero and solve would tell you when the profit was 0, not a maximum

Answer 8

@satellite73 yes it is a -5
but the question is asking to find the maximum profit

Answer 9

From P(x) = -5x^2 + 400x - 2550, we see that a = -5 and b = 400


Plug these values into
h = -b/(2a)
to get


h = -b/(2a)
h = -400/(2*(-5))
h = ______ (fill in the blank)

Answer 10

@jim_thompson5910 h=40

Answer 11

correct

Answer 12

so the x coordinate of the vertex is x = 40

plug x = 40 into the P(x) function to find the y coordinate of the vertex

Answer 13

P(x) = -5x^2 + 400x - 2550
P(40) = -5(40)^2 + 400(40) - 2550
= -8,000+16,000 -2550
= 5450
@jim_thompson5910 like this?

Answer 14

correct

Answer 15

x is in thousands, so is P(x)

so x = 40 means 40,000 dollars is spent on ads yielding a profit of 5450*1000 = 5,450,000 dollars, which is the max profit