Question: I am doing this math question and i am really confused on how to approach it.

This is the question:

A retired mathematics professor has decided to raise a goat. He owns a silo and a barn. The barns front wall is tangent to the silo at the corner. The silo has a circular base with a radius of 10 feet. The professor has decided to tether the goat to a chain that is anchored at the corner of the barn, the point of tangency. He has also cut the chain so that it is long enough to wrap around the silo exactly once- that is, the length of the chain equals the circumference of the silo. The barn length is longer than the chain.

This is an image of the barn (square), the silo (circle) and the goat's grazing area.

[![The drawing.](https://i.stack.imgur.com/uPWsn.png)](https://i.stack.imgur.com/uPWsn.png)

The answer I got is -4762.48876, but area is positive, so i made it 4762.48876.

These are the steps i took to getting the answer I have:
[![The drawing.](https://i.imgur.com/yzoL7j5.jpg)](https://i.imgur.com/yzoL7j5.jpg)
[![The drawing.](https://i.imgur.com/lXvKYwr.jpg)](https://i.imgur.com/lXvKYwr.jpg)

$$x(\phi)= -R\sin\phi + \phi R\cos\phi, \qquad y(\phi) = R-R\cos\phi - \phi R\sin\phi$$

Answer: 
You can't; that matrix has no inverse.

On the other hand, you know that$$X.\begin{bmatrix}1\\0\\0\end{bmatrix}=\begin{bmatrix}0\\0\\1\end{bmatrix}$$and therefore the first column of $X$ will be $\left[\begin{smallmatrix}0\\0\\1\end{smallmatrix}\right]$. You can compute the other columns by the same method. You will get that$$X=\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}.$$
I don't know how to put it, but it's basically a permutation matrix:
$$X=\begin{pmatrix}0 & 1 & 0\\ 0 & 0 & 1 \\ 1 & 0 & 0\end{pmatrix}$$