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Math Help - maximum when x and r are equal to P/pi+4

  1. #1
    Junior Member Jaycs's Avatar
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    maximum when x and r are equal to P/pi+4

    a norman window is a semi-circle sermounted on top of a rectangle. assum Parimeter of the window is P, a constant. Show the area of window is maximum when both x and r are equal to P/pi+4.
    Show maximum area is 1/2p^2/pi+4

    Stuck I tried solving for P first bu I think it is incorrect.
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  2. #2
    A Plied Mathematician
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    I would try to write the perimeter and the area as functions of x and r. Question: is the rectangle really a square? You don't provide another dimension for it, so I'm kind of assuming you have a square.
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    well if I draw a line across the diameter of the semi-circle it looks more like it is setting on a square. But the book states its a rectangle (no dimensions given) but when solving for area I guess it would end up the same.
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    Let me ask this question: is the diameter equal to the width of the rectangle, and x is equal to the height?
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  5. #5
    Junior Member Jaycs's Avatar
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    correct.
    so I begin by solving for a area of a semi-circle which 1/2(pi r^2) and a rectangle are which is L * W. ?
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    Junior Member Jaycs's Avatar
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    i have a radius of 2' and a hieght of 4' on rectangle
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    A Plied Mathematician
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    Quote Originally Posted by Jaycs View Post
    correct.
    so I begin by solving for a area of a semi-circle which 1/2(pi r^2) and a rectangle are which is L * W. ?
    But you also know the width of the rectangle in terms of r, right?

    What's the perimeter?
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  8. #8
    Junior Member Jaycs's Avatar
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    parimeter =16
    so it would be 16/1/2pi 4
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    I don't think you know any numbers yet. Stick with variables for now. Can you write down the area as a function of r and x, and the perimeter as a function of r and x?
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  10. #10
    Junior Member Jaycs's Avatar
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    parimeter = r^2(x)


    Area = 1/2 pi r^2
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  11. #11
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    No, I don't think you have it yet. The perimeter is one-half of a circle (the semicircle part), plus three sides of the rectangle (don't count the border between the rectangle and the semi-circle).

    For the area, you've only computed the area of the semicircle. What's the area of the rectangle? That has to be added into your formula as well.

    So, what do you get?
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  12. #12
    Junior Member Jaycs's Avatar
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    p= 1/2 pi^2+2x+L =

    1/2 pi r^2+2x+r^2
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  13. #13
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    But you know what L is as a function of r, right? Eliminate the extra variable!

    I don't buy your area formula yet. What's the area of a rectangle?
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  14. #14
    Junior Member Jaycs's Avatar
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    area of rectangle is L * W

    1/2pi r^2+2xr^2
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  15. #15
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    Why is the r squared in the second term?
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