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

  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. Dick

    Dick 25,746
    Science Advisor
    Homework Helper

    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. 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. 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. Dick

    Dick 25,746
    Science Advisor
    Homework Helper

    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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