Chapter 4 - Section 4.1 - Angles and Radian Measure - Exercise Set - Page 534: 122

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.

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