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Math Help - Optimization

  1. #1
    Junior Member
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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 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

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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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