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Math Help - Norman window maximum problem

  1. #1
    DBA
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    Norman window maximum problem

    A Norman Window has the perimeter of 30ft, find the dimentions of the window so that the greatest possible amount of light is admitted.

    I did:

    Perimeter of half circle = pi*r
    Perimeter of the rectangle (3sides) 2r + 2y

    Perimeter of the window

    30 = pi*r + 2y + 2r

    Solve for y

    y = 15 - r - (pi*r)/2


    I understand out of the text that I need the max area of the window:

    Area half circle = (pi*r^2)/2
    Area of rectangle = 2r * y

    Area of the window A= (pi*r^2)/2 + 2r*y

    Now I plug in y

    A= (pi*r^2)/2 + 2r* [15 - r - (pi*r)/2]


    My question:
    Can I write y = 15 - r - (pi*r)/2 so that I can combine the two terms with r in one term?
    Please, including the way you did that.

    And what is the derivative then? A'(r)?

    Please, do not give me any links of similar problems, I looked at them but this did not help me with my problem.

    Thanks.
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  2. #2
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    Yes, you may substitute 15-r-\frac{\pi r}{2} for y, as an increase in r implies a corresponding decrease in y, and the area function A may therefore be written in terms of r alone.
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  3. #3
    DBA
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    Hi,
    I meant if I can write y = 15-r-(pi*r)/2 so that there is only one term of r in y not in the area formula.
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  4. #4
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    Yes, you may factor out r:

    y=15-r\left(1+\frac{\pi}{2}\right).
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