An athletic field with a perimeter of 0.25 miles consists of a rectangle with a semicircle at each end, as shown below. Find the dimensions that yield the greatest possible area for the rectangular region.
This is the work that I did below. I was wondering if this was the greatest possible area for the rectangle below.
Near the end of page $1$, you wrote $r=\frac{2}{16\pi}$ when you meant to say $2r=\frac{2}{16\pi}$
Once we found out that $r=\frac{1}{16\pi}$, we can compute $$l=\frac18 - \pi r= \frac18 -\frac1{16}=\frac1{16}$$ directly without finding $A$ explicitly.