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Optimization of a rectangular window surmounted on a semicircle

  1. Dec 8, 2012 #1
    1. The problem statement, all variables and given/known data
    A decorative window has the form of a rectangle surmounted by a semicircle whose diameter is equal to the top of the rectangle. If the TOTAL perimeter of the window 16+pi, then what is the maximum area?

    A. 25.653
    B. 32.148
    C. 15.923
    D. 38.047
    E. 30.018

    Correct answer is A: 25.653, but explain step by step please.

    2. The attempt at a solution

    I completely started off on the wrong foot here.
    What I did was made the radius = x/2 where x is the total width/diameter of the rectangle/circle. Then I made the equations:

    P=2∏(x/2)+2x+2y=16+pi
    A=∏(x/2)^2+xy=z

    I seem to not be getting the answer after I plug everything in, so I know im starting off wrong.
    Please explain by a step-step process.
    Thanks in advance.
     
    Last edited: Dec 8, 2012
  2. jcsd
  3. Dec 8, 2012 #2

    SteamKing

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    Homework Helper

    Eliminate some of your unknowns by assuming that the rectangle has a certain ratio of width to height, so that x = r*y. Then the perimeter can be expressed as a function of x, which can be substituted into the area formula.
     
  4. Dec 8, 2012 #3

    Ray Vickson

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    Your formulas for P and A are wrong. Draw a carefully-labelled diagram, showing x, y, x/2, etc., and then see where your error lies.
     
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