(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the dimensions of the largest rectangle with sides parallel to the axes that can be inscribed in the ellipse x^2 + 4y^2 = 4

2. Relevant equations

3. The attempt at a solution

I simplified the equation of the ellipse into the ellipse formula:

x^2/4 + y^2 = 1

Then I manipulated the equation to isolate y:

y = (sqrt(4-x^2)/2)

Then, since the area of the rectangle can be divided into four equal parts with equal length and width, I substituted my y into my area formula, a = 4xy:

a = 2x(sqrt(4-x^2))

Now I find the derivative of my area...

a' = (8-4x^2)/sqrt(4-x^2)

... Set it equal to 0 to find my maximum value for x:

(4(2-x^2))/sqrt(4-x^2) = 0

And find that x is equal to sqrt(2). (which is the right answer, now I just need y)

Then, I substituted my value for x back into my area formula (a = 4x(sqrt(4-x^2)/2)))

4(sqrt(2))(sqrt(4-(sqrt(2))^2)/2)

and end up with my area as 4.

Then I substituted both this area and my x value back into my area formula, a = 4xy:

4 = 4(sqrt(2))y

and find that y is equal to 4. However, the back of my book tells me that I should have y = 2sqrt(2).

I have looked it over several times and cannot find my mistake. Help would be much appreciated!!

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# Rectangle inscribed in ellipse

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