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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.