# Thread: How Do You Find The Maximum Area Of A Rectangle In An Eclipse?

1. ## How Do You Find The Maximum Area Of A Rectangle In An Eclipse?

(deleted)

2. Originally Posted by AlphaRock
Can somebody show me step by step how to get the maximum area of a rectangle in an eclipse? Because I'm burnt out.

$\displaystyle (x^2)/9 + (y^2)/4 = 1$

Thanks!
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.