Optimization of Area; Inscribed Rectangles

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The discussion focuses on optimizing the area of an inscribed rectangle within a semicircle. The area is expressed as 2xy, with y being substituted from the semicircle's equation. Differentiation with respect to x identifies critical points, leading to the conclusion that x = sqrt(6) maximizes the area. The resulting dimensions of the rectangle are determined to be 2sqrt(6) for the base and sqrt(6) for the height. This approach effectively finds the optimal dimensions for the inscribed rectangle.
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Homework Statement



http://i.minus.com/jbkHm5oH1LfQ1k.png

Homework Equations



The area of a rectangle is its base times its height.

The Attempt at a Solution



The rectangle is inscribed. Its area is 2xy. I can substitute in the equation of the semicircle to get rid of the y-term in my area equation. I can then differentiate with respect to x and find what zeros the derivative (or causes the derivative not to exist) and test the points I find that are on the domain of x (0 to square root of 12).

I find x = sqrt6 as that point that maximizes area and therefore the dimensions must be 2sqrt6 and sqrt6.

http://i.minus.com/jFWGaWoQsb5FX.jpg
 
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