optimization: Find dimensions of rectangle of maximum area,

[degreeplus]

Maximum Area: Find the dimensions of the rectangle of maximum area, with sides parallel to the coordinate axes, that can be inscribed in the ellipse given by

(x^2/144)+(y^2/16)=1

I have the equation:

A=(2x)(2y)

and used the equation y=4-(4x/12) to substitute for y.

Then I found the derivative to be

A'=16-(8/3)x
where the critical number turns out to be x=6

then I found the dimensions as 12 by 4 but this doesn't make sense to me. I think there is something wrong with my initial equation and need help finding out what I need to use.

I dont have the answer so I can't be sure if Im solving this correctly.

Another equation I think about is:
A=2(x-a)*2(y-b) where a=16 and b=4
A'=-(8/3)x-16
critical #'s
x=-6
and my dimensions of rectangle turn out to be 12 by 12 but i dont think this is correct either.

I could really use some help on what the initial equation should be.

 

[galactus]

You're getting there, but a little off.

Let's derive the general case and then use your a and b.

Ellipse equation: \(\displaystyle \L\\\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\)

Let's break your rectangle up into 4 smaller rectangles with one in each quadrant, each having area xy.

That way the entire rectangle will have area 4xy.

Solving the ellipse equation for y gives us \(\displaystyle \L\\y=\frac{b}{a}\sqrt{a^{2}-x^{2}}\)

Sub that into 4xy, getting \(\displaystyle \L\\A=4x(\frac{b}{a}\sqrt{a^{2}-x^{2}})\)

Now, differentiate: \(\displaystyle \L\\\frac{dA}{dx}=\frac{4b\sqrt{a^{2}-x^{2}}}{a}-\frac{4bx^{2}}{a\sqrt{a^{2}-x^{2}}\)

Set to 0 and solve for x:

\(\displaystyle \L\\\frac{4ba^{2}-8bx^{2}}{a\sqrt{a^{2}-x^{2}}}=0\)

\(\displaystyle \L\\4ba^{2}=8bx^{2}\)

\(\displaystyle \L\\x=\frac{a}{\sqrt{2}}\)

Go ahead and find y now that you have x. You can use your a and b values to find the area of the rectangle of largest area which will fit in your ellipse.

There is a ratio of \(\displaystyle \frac{\pi}{2}\) regarding the rectangle of max area and the area of the entire ellipse.