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March 29, 2017

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Building windows: Window World, Inc., is responsible for designing new windows for the expansion of the campus chapel. The current design is shown in the figure. The metal trim used to secure the perimeter of the frame is 126″ long. If the maximum window area is desired (to let in the most sunlight), what will be (a) the dimensions of the rectangular portion of each window? (b) the total area of each window

  • Algebra - ,

    no figure, but from the context it appears to be a rectangle and something else. Since it's a church, and the same kind of problem has appeared here before, I assume the top portion is a semi-circle.

    So, with width w and height h, the perimeter is w+2h+pi*w/2. The trim is not used between the top and bottom portions of the window.

    So, we have

    w+2h+pi*w/2 = 126
    so,
    h = 63 - (pi+2)w/4

    window area is rectangle + half-circle

    a = wh + pi/8 w^2
    = w(63 - (pi+2)w/4) + pi/8 w^2
    = 63w - pi/8 (2pi+3)w^2

    da/dw = 63 - pi/4 (2pi+3)w
    max/min area is where da/dw = 0

    w = 252/(3pi + 2pi^2) = 8.64

    Now you can figure the areas.

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