Thread: 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

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:



and let x=k be the coordinate from the center to the end on the right.
The y coordinate in the first quadrant is .

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:



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



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?

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.