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Thread: Optimization

  1. #1
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    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!
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  2. #2
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    Hello, kdogg121!

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

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

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

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


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

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

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

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


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

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