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- Mar 26th 2009, 06:35 PMAlphaRockHow Do You Find The Maximum Area Of A Rectangle In An Eclipse?
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- Mar 26th 2009, 07:14 PMmollymcf2009
Area of a rectangle =$\displaystyle xy$

Equation of the ellipse = $\displaystyle \frac{x^2}{9} + \frac{y^2}{4} = 1$

First solve your ellipse equation for x.

$\displaystyle \frac{4x^2 + 9y^2}{36} = 1

$

$\displaystyle 4x^2 + 9y^2 = 36$

....

$\displaystyle x = ^{+}_{-} \sqrt{\frac{36-9y^2}{4}}$

Plug this into your area equation:

$\displaystyle A(y) = (\sqrt{\frac{36-9y^2}{4}})y

$

Take your derivative and put it = to zero. Solve. Then just plug in your value for y into the ellipse equation that you solved for x above and this will give you your dimensions of your rectangle with the largest area. Then just use your values to find an exact area.