My Math Forum Rectangle and Ellipse and Area

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 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 $x^2/a^2 + y^2/b^2= 1$ $a , b > 0$ 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 $y=\pm\frac{b}{a}\sqrt{a^{2}-x^{2}}$ Sub this in the simple little area equation: $A=4x(\frac{b}{a}\sqrt{a^{2}-x^{2}})$ Differentiate, set to 0 and solve for x: $\frac{dA}{dx}=\frac{4b}{a}\sqrt{a^{2}-x^{2}}-\frac{4bx^{2}}{a\sqrt{a^{2}-x^{2}}}=0$ $=\frac{4b(a^{2}-x^{2})-4bx^{2}}{a\sqrt{a^{2}-x^{2}}}=0$ $\Rightarrow 4ba^{2}-4bx^{2}=4bx^{2}$ $x=\frac{a}{\sqrt{2}}$ The width is $\frac{2a}{\sqrt{2}}$ 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 $\left(\frac{2a}{\sqrt{2}}\right)\left(\frac{2b}{\s qrt{2}}\right)=\fbox{2ab}$ 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 $x=a\cdot cos(t), \;\ y=b\cdot sin(t)$ The area of said rectangle is $A(t)=4abcos(t)sin(t)$ $\frac{dA}{dt}=8abcos^{2}(t)-4ab$ Doing the same, setting to 0 and finding t, we get $t=2C{\pi}+\frac{\pi}{4}$. When C=0, we have $t=\frac{\pi}{4}$ $a\cdot cos(\frac{\pi}{4})=\frac{a}{\sqrt{2}}, \;\ b\cdot sin(\frac{\pi}{4})=\frac{b}{\sqrt{2}}$ Just as before.

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# atea of greatest rectangle inscribed in ellipse

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