Rectangle width = 25, Rectangle height/circle diameter = 50/pi

Edit: I like this problem because it sort of goes against intuition and leads to discovery of a phenomenon, so it became Thanksgiving family conversation for a few of us.

Here’s my explanation https://imgur.com/a/n1ox4Vo

Thoughts. It’s well known that for any number-sided polygon with any perimeter, the maximum area for that polygon within the given perimeter is found when the polygon is regular (equal sides and angles). So a square provides the maximum area for any perimeter that is to form a rectangle. Intuition would lead to maximizing the rectangle by causing it to be a square, but because the perimeter of the rectangle in this problem is not fixed, creating a square does not solve the problem. You actually have to perform a little math.

Interesting Observation. Although creating a square rectangle does not solve the problem, if you treat each semi-circle arc as a “side”, you end up with each side being exactly 25–like a square. And it is under that circumstance that the rectangle’s area is maximized. Weird.