# Math Help - Word problem

1. ## Word problem

Hi,

I need some assistance with the following word problem :

The profit $P$ made by a cinema from selling $x$ bags of popcorn can be modeled by:

$P = 2.36x - \frac{x^2}{25000} - 3500, 0 < x < 50000$

NOTE: the $<$ is suppose to be also "or equal to" i just didn't know the syntax.

I need to:

a - Find the intervals on which [tex]P[/math is increasing and decreasing.

b - If you owned the cinema, what price would you charge to obtain a maximum profit for popcorn? Explain please.

2. solve $\frac {dP}{dx} = 0$ to find the critical points

3. Right so I did this:

$P'=2.36-\frac{1}{12500}x = 0$
$x = 29500$

and I tested in the intervals : (- infinity, 29500) , (29500, infinity) and found its increasing on the first and decreasing on the second (respectively).

now how can I answer part b of my question ?

4. If the profit value is increasing up to a certain x-value, and then decreasing after that x-value, where then would be the maximum value of P?