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Math Help - express the area A as a function of the width X

  1. #1
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    express the area A as a function of the width X

    A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 22 ft, express the area A of the window as a function of the width x of the window.

    I will label 2 sides as y and base as x
    π is pi

    Perimeter = 2y + x + ((2πr)/2)

    since there isn't really a base (x) at the top of the window I will remove x from the equation above.

    I will then plug in Radius as (x/2)

    p= 2y + ((2π(x/2))/2)
    = 2y + (πx)/2

    22 = 2y + (πx)/2

    Solving for y in above equation I came up with:

    y = 11 - [(πx)/4]




    Now for the Area, A(x)

    A(x) = xy + (πr^2)/2


    Plugging in Y and Radius obtained before my answer to this problem came to:


    A(x) = x[11 - ((πx)/4)] + [(π (x/2)^2)/2]


    any thoughts?
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  2. #2
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    Re: express the area A as a function of the width X

    Quote Originally Posted by NeoSonata View Post
    A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 22 ft, express the area A of the window as a function of the width x of the window.

    I will label 2 sides as y and base as x
    π is pi

    Perimeter = 2y + x + ((2πr)/2)

    since there isn't really a base (x) at the top of the window I will remove x from the equation above.

    I will then plug in Radius as (x/2)

    p= 2y + ((2π(x/2))/2)
    = 2y + (πx)/2

    22 = 2y + (πx)/2

    Solving for y in above equation I came up with:

    y = 11 - [(πx)/4]




    Now for the Area, A(x)

    A(x) = xy + (πr^2)/2


    Plugging in Y and Radius obtained before my answer to this problem came to:


    A(x) = x[11 - ((πx)/4)] + [(π (x/2)^2)/2]


    any thoughts?
    All your calculations are OK.

    Now expand the bracket and collect like terms:

    A(x)= 11x - \frac \pi4 x^2 + \frac \pi8 x^2=11x-\frac \pi8 x^2

    ... that's all!
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