Answer
The expression for the area enclosed by the track as a function of radius $r$ of the track is $A\left( r \right)=440r$.
Work Step by Step
The total length of the path would be the sum of the length of straight paths and circumference of two semi circles.
Write the expression for the total length of track.
$2x+2\left( \pi r \right)=880$
Calculate $x$ in terms of r.
$\begin{align}
& 2x+2\left( \pi r \right)=880 \\
& 2x+2\pi r=880 \\
& 2x=880-2\pi r \\
& x=\frac{880-2\pi r}{2}
\end{align}$
Solve further,
$\begin{align}
& x=\frac{2\left( 440-\pi r \right)}{2} \\
& =440-\pi r
\end{align}$
The area enclosed by the track is the sum of the area of rectangle of sides x and $2r$ and semicircle of radius $r$.
Write the expression of the area enclosed by the track.
$\begin{align}
& A=x\left( r \right)+2\left( \frac{1}{2}\pi {{r}^{2}} \right) \\
& =xr+\pi {{r}^{2}}
\end{align}$
Substitute $440-\pi r$ for $x$.
$\begin{align}
& A=\left( 440-\pi r \right)r+\pi {{r}^{2}} \\
& =440r-\pi {{r}^{2}}+\pi {{r}^{2}} \\
& =440r
\end{align}$
The area is function of r only which can also be expressed as,
$A\left( r \right)=440r$
Hence, expression for the area enclosed by the track as a function of radius $r$ of the track
is $A\left( r \right)=440r$.
