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Math Help - Maximization Problem Help

  1. #1
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    Maximization Problem Help

    Here's my question...

    A Norman window, one that is formed by placing a semicircle on top of a rectangle, still remains a popular architectural feature. If the perimeter of the window is 300cm, find the radius of the semicircle that will maximize the window's area (let in the most light).

    I came up with this equation... 300 = 2x + y + ((pi*y)/2)
    x = rectangle height
    y = rectangle width

    These questions really bug me and I'm really stuck! Help please!
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  2. #2
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    Quote Originally Posted by lionpants View Post
    Here's my question...

    A Norman window, one that is formed by placing a semicircle on top of a rectangle, still remains a popular architectural feature. If the perimeter of the window is 300cm, find the radius of the semicircle that will maximize the window's area (let in the most light).

    I came up with this equation... 300 = 2x + y + ((pi*y)/2)
    x = rectangle height
    y = rectangle width

    These questions really bug me and I'm really stuck! Help please!
    First I'd let the width of the rectangle be 2y so that the radius of the semi-circle is y.

    Then 2x + 2y + pi y = 300 and you can re-arrange this to make x the subject.

    A = 2xy + pi y^2 and you can sub x in terms of y from the above.

    So you have A as a function of y.

    Now solve dA/dy = 0 for y and test that this corresponds to a maximum.
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