Question

A cosmetic company sells nail polish at $5 a bottle. The cost function is given as
C(x) = -0.03x² + 9.215x + 550 where C is the cost of producing x bottles.
How many bottles should the company sell per month in order to make a profit?

How do to?

Answer 1

I set it equal to 5x getting C(x) = -0.03x² + 4.215x + 550 but now what?

Answer 2

The break even point would be the x-intercepts but one of them is in the negatives... so I was confused

Answer 3

Can they make a negative number of bottles?

Answer 4

But then how else can we find when the start making a profit?

Answer 5

"break even point would be the x-intercepts but ONE of them is in the negatives"

Answer 6

the other one is not a negative so I would just say they need to sell less than... 222.8?

Answer 7

If they sell less than 222.8, are they in the positive for profit (making money), or negative?

Answer 8

The revenue is 5x. Whenever the revenue is greater than the cost they will make a profit. So write: 5x>-.03x^2+9.25x+550 and solve

Answer 9

@Mertsj yea... I never thought of that

Answer 10

@agent0smith so saying less than 222.8 is wrong?

Answer 11

It'd be easier/make more sense to find the profit function, which is revenue - cost

The way you did it's backwards... cost - revenue.

Answer 12

P(x) = R(x) - C(x)

for profit, then just find where it's greater than zero.

Answer 13

yea thats why I moved the 5x on the right side and set the equation equal to zero. or do you mean something else?

Answer 14

or is it 0 < -0.03x² + 9.215x + 550 ? @agent0smith

Answer 15

P(x) = 5x - ( -0.03x² + 9.215x + 550)

Then you need where P(x) > 0

Answer 16

Thankyou so much sir

Answer 17

That way it makes more sense... if you do it the other way around, you have to remember where it's negative is the region you want.

Answer 18

but with that way the parabola has a minimum point but should have a max?

Answer 19

when i subtract 5x - the cost function

Answer 20

If your functions and such in the original post are correct, then that appears correct. The cost function has a limit though, after they produce 358 ish bottles, cost becomes negative.

Answer 21

I have


0.03x^2-4.215-550 > 0

is that correct

Answer 22

0.03x^2-4.215x-550 > 0

Answer 23

Looks right

Answer 24

now how can i find the number of bottles

Answer 25

You already found it earlier. x>222.8

Answer 26

alright

Answer 27

Since that's where profit goes into the positive region.