Maximum Area

• Jul 5th 2009, 05:52 PM
McScruffy
Maximum Area
I'm having some trouble developing an equation to differentiate for this problem. Any help would be appreciated.

Consider a symmetric cross inscribed in a circle with radius r. Determine the value x, where x is the length from the center of the cross to one of its ends, such that the value of x maximizes the area of the cross.

Thanks
• Jul 5th 2009, 06:07 PM
apcalculus
Quote:

Originally Posted by McScruffy
I'm having some trouble developing an equation to differentiate for this problem. Any help would be appreciated.

Consider a symmetric cross inscribed in a circle with radius r. Determine the value x, where x is the length from the center of the cross to one of its ends, such that the value of x maximizes the area of the cross.

Thanks

You need an area function in terms of x only.
Let the equation of the circle be:

$\displaystyle x^2 + y^2 = r^2$

and let x=k be the coordinate from the center to the end on the right.
The y coordinate in the first quadrant is $\displaystyle y =\sqrt{r^2 - k^2}$.

Split the cross into two parts:

The horizontal band including the shared square with the vertical band.
This horizontal band has length 2k and height 2y, so its area is:

$\displaystyle A = 4 y k = 4 \sqrt{r^2-k^2} k$

The upper half of the vertical band has area given by 2y*height where height is k - y.

$\displaystyle B = 2y (k-y) = 2 \sqrt{r^2-k^2} (k - \sqrt{r^2-k^2})$

Combine A and 2B (note: it's 2B because you need the lower half of the vertical band as well), then do calculus with it.

Does this help?
• Jul 5th 2009, 06:25 PM
McScruffy
Yes that helps a great deal. That's kind of the idea that I was thinking about, I just couldn't seem to get it worked out into a reasonable equation. But that makes a lot of sense.
Thanks.