June 16th, 2009, 11:33 AM  #1 
Newbie Joined: Jun 2009 Posts: 3 Thanks: 0  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. 
June 16th, 2009, 12:10 PM  #2 
Guest Joined: Posts: n/a Thanks:  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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area, ellipse, rectangle 
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