Answer
The required solution is $286\ \text{mi}$
Work Step by Step
The angle $\theta $ for the change in direction is $20{}^\circ $.
Then, convert it into radians:
$\begin{align}
& \theta =20{}^\circ \\
& =20{}^\circ \left( \frac{\pi }{180{}^\circ } \right) \\
& =\frac{\pi }{9}
\end{align}$
And the curve distance $s$ is $100$ miles.
The arc’s length subtended by an angle at the center of the circle is given by
$s=r\theta $
Here, $r$ is the radius of the railroad curve.
Rearrange for $r$:
$r=\frac{s}{\theta }$
Put $100$ miles for $s$ and $\frac{\pi }{9}$ for $\theta $:
$\begin{align}
& r=\frac{100\ \text{mi}}{\frac{\pi }{9}} \\
& =\frac{900\ \text{mi}}{\pi }
\end{align}$
Put $\pi =3.14159$:
$\begin{align}
& r=\frac{900\ \text{mi}}{3.14159} \\
& =286.47\ \text{mi}\approx 286\ \text{mi}
\end{align}$
Hence, the radius of the railroad curve is $286$ miles.
