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June 16th, 2009, 11:33 AM   #1
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Rectangle and Ellipse and Area

We have a Rectangle with sides parallel to axes (x and y). The rectangle is inside of the Ellipse



Calculate the dimensions of the rectangle with maximum area.
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June 16th, 2009, 12:10 PM   #2
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Re: Rectangle and Ellipse and Area

Draw a general ellipse and then draw a rectangle inside to get a picture.

Center everything at the origin. The width of the rectangle is 2x and the height is 2y. It's area would be A=4xy.

Now, solve the ellipse equation for y. We get

Sub this in the simple little area equation:



Differentiate, set to 0 and solve for x:









The width is

The height can be found by subbing x back in the above and finding it is the same.

Therefore, the rectangle of max area is a square. Which is normally the case in these instances.

The max area is then

There are various ways to tackle this. You could represent the ellipse in parametric perhaps and go that route. Use a little trig maybe.

Now, try this one. Find the ellipsoid of max volume that can be inscribed inside a cone of radius R and height H.

Parametrically, the ellipse has equation

The area of said rectangle is



Doing the same, setting to 0 and finding t, we get .

When C=0, we have



Just as before.
 
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