Simple Calculus Word Problem (using derivatives to anaylze function models)

  1. Feb 27, 2008 #1
    Hello, new here, first post. Just need some help with homework.

    Question One

    1. The problem statement, all variables and given/known data

    This norman window is made up of a semicircle and a rectangle. The total perimeter of the window is 16 cm. What is the maximum area?

    **
    * * <<< Semicircle
    *****
    | | <<< Rectangle
    L | |
    ______
    D
    2. Relevant equations

    P (total) = 2L + D + (pi * d)

    A (total) = D * L + (pi(d/2)^2)/2)


    3. The attempt at a solution

    What I did was using this equation:
    16 = 2L + D + ((pi * d)/2)
    L = 8 - d/2 - ((pi * d)/4)

    A = D (8 - d/2 - ((pi * d)/4)) + (pi (d/2)^2)/2)
    A = 8d - (d^2)/2
    A' = 8 - d
    Let 0 = A' to find critical value
    then 8 = d.

    When I sub that back into the original equation, I get L as a value less than 8, which doesn't make sense. (I think it works out to be L = 4 - pi)

    I'm pretty much lost, sorry if this is too messy to read, any help would be appreciated. Thanks
     
  2. jcsd
  3. Feb 27, 2008 #2

    Dick

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    You made a great start. But where did you get "16 = 2L + D + ((pi * d)/2)"?? The "/2" wasn't in your original expression for P. You are just making algebraic mistakes.
     
  4. Feb 27, 2008 #3
    I think you're doing fine up until this point:

    A = D (8 - d/2 - ((pi * d)/4)) + (pi (d/2)^2)/2)

    Which should simplify into

    [tex]

    A = 8*D - \frac{D^2}{2} - \frac{pi*D^2}{4} + \frac{pi*D^2}{8}
    [/tex]

    You would then go on to take the derivate and then set it to zero and solve for your D value

    I've been beaten =(
     
    Last edited: Feb 27, 2008
  5. Feb 27, 2008 #4
    ah sorry, its actually supposed to be "/2", that way its half the area, sorry the drawing didnt show up. its supposed to be a semi-circle connected to a rectangle.

    sorry, that was a typing error as well haha.

    A = L * D + (pi*d)/2
    which becomes

    A = 8 - d/2 - ((pi * d)/4)

    this still doesnt work...I think i'm using the wrong equations somehow
     
  6. Feb 27, 2008 #5

    Dick

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    Why isn't there a D in all of the terms of A? I think you understand this problem perfectly well and you are using the right equations. You are simply making typographical mistakes right and left. Get a clean sheet of paper, calm down and take a stress pill and you can do this.
     
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