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Thread: Max/Min Problem

  1. #1
    Jan 2010

    Max/Min Problem

    Hey, I really need help solving this problem. Help would be greatly appreciated.

    1. A rectangle is to be inscribed in a semicircle with radius 4, with one side on the semicircle's diameter. What is the largest area this rectangle can have?

    This problem was under the "Maxima and Minima" section of a math book, but I can't seem to figure out how to solve it.
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  2. #2
    May 2008
    I always find drawing it first to get better idea of the problem is a good start.

    Make the length of the rectangle 2x. Look at the attached picture and you can see the length from centre of rectangle to top corner is the radius; 4. The length from centre to bottom corner is x. So height of rectangle will be $\displaystyle \sqrt{16-x^2}$ (using pythagoras).

    Area of rectangle is $\displaystyle A = 2x (\sqrt{16-x^2})$
    differentiate that and solve for x to get the answer. You should get two answers, one of which will be zero and so obviously not the max answer!
    hint: To make it easier I'd square both sides: $\displaystyle A^2 = 4x^2 (16-x^2) = 64x^2 - 4x^4$
    Attached Thumbnails Attached Thumbnails Max/Min Problem-semi-circle-rectangle.jpg  
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