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Math Help - optimization problem

  1. #1
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    optimization problem

    Determine the area of the largerst rectangle that can be inscribed in a right cemicircle with a radius of 10 units. Place the length of the rectangle along the diameter.
    Thank you in advance!
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  2. #2
    Senior Member apcalculus's Avatar
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    Quote Originally Posted by anna12345 View Post
    Determine the area of the largerst rectangle that can be inscribed in a right cemicircle with a radius of 10 units. Place the length of the rectangle along the diameter.
    Thank you in advance!

    Start with a picture, with the semi-circle centered at the origin. Let top right corner of the rectangle have coordinates (x, y) where both x and y are positive (first quadrant).

    x^2 + y^2 = 100 since the point is located on the circle

    The area is A = (2x) * y = 2xy

    Substitute into the area function so it depends on x or y, only. Then differentiate the area function to find either dA/dx or dA/dy. Then solve for critical numbers. You may want to verify that your solution is a max (and not a min) using one of the derivative tests.

    Hope this helps!
    Last edited by apcalculus; April 14th 2009 at 09:36 AM. Reason: spell error
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