Determine all real numbers $a$, where $a>0$, for which there exists a nonnegative continuous function $f(x)$, defined on $[0,a]$ with the property that the region $$R=\big\{(x,y)\,\big|\, 0\le x\le a\text{ and } 0\le y\le f(x)\big\}$$ has perimeter $k$ units and area $k$ units for some real number of $k$.
I apologize for not yet knowing how to type equations as they are seen in most questions on this forum. I joined about 15 minutes ago.