1. ## Optimization

I am having trouble with an Optimization problem, I can find everything except for the maximized amount. It wants me to determine the level of production q where P(q) is maximized. Here is the given information:

p(q)= 180-2q ; C(q)= q^3+5q+162

I found R(q) Revenue to be: 180q-2q^2 I then need to subtract c(q) from that which looks like this: (180-2q^2) - (q^3+5q+162) and I got P(q) to be:
-q^3-4q+175q-162=P(q) To find the maximized value you are supposed to take the derivative of P(q) which I got to be: -3q^2-4q+175 to find the maximized level of production you are supposed to solve for q, however I am confused how to do this because there is a q^2 term there. The book's answer for maximized production is q=7 Any help would be appreciated, thanks!

2. Hello, kdogg121!

Determine the level of production [tex]q where $P(q)$ is maximized.

Here is the given information: . $p(q)\:= \:180-2q,\quad c(q)\:= \:q^3+5q+162$

I found Revenue to be: . $R(q)\:=\:180q-2q^2$
I then need to subtract $c(q)$ from that which looks like this:
$(180-2q^2) - (q^3+5q+162)$
and I got: . $P(q)\:=\:-q^3-4q+175q-162$

To find the maximized value, take the derivative of $P(q)$
which I got to be: $P\,'(q)\:=\: -3q^2-4q+175$
To find the maximized level of production you are supposed to solve for $q,$
. .
All of this is correct!
Of course, we solve: . $P\,'(q) {\bf{\color{blue}= 0}}$

We have: . $-3q^2 - 4q + 175 \:=\:0$

Multiply by -1: . $3q^2 + 4q - 175\:=\:0$ . . .
You know how to solve a quadratic, right?

Factor: . $(y - 7)(3y + 25) \:=\:0$

. . and we get: . $q \:=\:7, -\frac{25}{3}$

And the positive root is: . $q = 7$

3. Thank You, I forgot about the algebra, now it's easy.