A railroad curve is to be laid out on a circle. What radius should be used if the track is to change direction by \(25^\circ\) in a distance of \(40\) meters?
Solution:
We know:
\[ \text{Arc Length} = r\theta \]
Given:
\[ s=40 \text{ m} \]
\[ \theta=25^\circ \]
Convert angle into radians:
\[ 25^\circ \times \frac{\pi}{180} = \frac{5\pi}{36} \]
Using,
\[ s=r\theta \]
\[ 40=r\times\frac{5\pi}{36} \]
\[ r=\frac{40\times36}{5\pi} \]
\[ r=\frac{288}{\pi} \]
\[ r\approx91.67 \text{ m} \]
Therefore, the required radius is:
\[ \boxed{91.67 \text{ meters (approximately)}} \]