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Math Help - How Do You Find The Maximum Area Of A Rectangle In An Eclipse?

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

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    Last edited by AlphaRock; April 1st 2009 at 01:44 AM.
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    Senior Member mollymcf2009's Avatar
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    Quote Originally Posted by AlphaRock View Post
    Can somebody show me step by step how to get the maximum area of a rectangle in an eclipse? Because I'm burnt out.

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

    Thanks!
    Area of a rectangle = xy

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

    First solve your ellipse equation for x.

    \frac{4x^2 + 9y^2}{36} = 1<br /> <br />
    4x^2 + 9y^2 = 36

    ....

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

    Plug this into your area equation:

    A(y) = (\sqrt{\frac{36-9y^2}{4}})y<br />
    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.
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