Question

A railroad curve is to be laid out on a circle. What radius should be used if the track is to change direction \[{{25}^{\circ }}\] in the direction of 40 meters?

Answer 1

Hint: For the above question first of all we will change the angle into radians from degrees by using the conversion \[{{1}^{\circ }}=\dfrac{\pi }{180}radians\].
Then we will use the formula as given below:
\[r=\dfrac{l}{\theta }\], where ‘r’ is the radius of the circle, ‘l’ is the length of arc and \['\theta '\] is the angle subtended by the arc of length at center. Also, \['\theta '\] must be in radian.
Complete step-by-step answer:
We have been given that a train is to change direction by \[{{25}^{\circ }}\] in a direction of 40 meters on a circular path. Now we know that \[{{1}^{\circ }}=\dfrac{\pi }{180}radians\].
\[\Rightarrow {{25}^{\circ }}=\dfrac{\pi }{180}\times 25=\dfrac{5\pi }{36}radians\]

We know that on a circular path, if the length of arc is ‘l’, the radius of the circle is ‘r’ and the angle subtended by the arc at center is \['\theta '\], then \[r=\dfrac{l}{\theta }\], where \['\theta '\] must be in radians.
Now we have l = 40 meters and \[\theta =\dfrac{5\pi }{36}radians\], then:
\[\begin{align}
  & \Rightarrow r=\dfrac{l}{\theta } \\
 & \Rightarrow \dfrac{40}{{}^{5\pi }/{}_{36}}=\dfrac{40\times 36}{5\pi }=\dfrac{288}{\pi } \\
\end{align}\]
Since \[\pi =\dfrac{22}{7}\], \[\Rightarrow r=\dfrac{288\times 7}{22}=91.63m(approx)\]
Therefore, the radius is equal to 91.63 meters (approximately).

Note: Be careful while changing the radians into degree and don’t get confused in the conversion 1 radian = \[\dfrac{180}{\pi }\] degrees because in a hurry, we might end up taking 1 radian = \[\dfrac{\pi }{180}\] degrees which is absolutely wrong. Also remember that in \[r=\dfrac{l}{\theta }\], \[\theta \] must be in radians. We cannot substitute the value in degree or other units of angle.