Question

A train 576 m long crosses a tunnel in 1 min 3 seconds. Find the length of the tunnel if the speed of the train is 48 km/hr?

Answer 1

Hint: Let us assume that the length of the tunnel is “t”. We know that the formula which contains speed, distance and time i.e. $\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}$. Now, the total distance that train has covered is the addition of the length of tunnel and the length of the train, time is given as 1 min 3 seconds (convert this time in seconds) and speed is given as 48 km/hr so convert this speed into m/sec. Substitute these values in the speed formula and solve the equation will give you the value of “t”.

Complete step-by-step answer:
It is given that the length of the train is 576 m and time it takes to cross the tunnel is 1 min 30 seconds with a speed of 48 km/hr.
Let us assume that length of the tunnel as “t”.
We have given speed, distance and time so we know the relation between speed, distance and time as:
$\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}$……….. Eq. (1)
Converting the units of speed in m/sec by multiplying 48 by 1000 and dividing this result by 3600 we get,
$\begin{align}
  & 48\left( \dfrac{1000}{3600} \right) \\
 & =48\left( \dfrac{10}{36} \right) \\
\end{align}$
We know that 48 is divisible by 6 by 8 times and 36 is divisible by 6 by 6 times.
$\begin{align}
  & 8\left( \dfrac{10}{6} \right) \\
 & =\dfrac{40}{3} \\
\end{align}$
From the above, we have calculated the speed as $\dfrac{40}{3}\text{m/sec}$
We will use the conversion, 1 minute = 60 seconds. Now, converting the time of 1 min 30 seconds in seconds by multiplying 1 min by 60 and adding this result to 30 seconds we get.
$\begin{align}
  & \left( 1\left( 60 \right)+30 \right)\sec \\
 & =90\sec \\
\end{align}$
The total length traversed by the train before completely crossing the tunnel is equal to the addition of length of the train with the length of the tunnel.
$576+t$
Now, substituting the values of speed, distance and time that we have just calculated in eq. (1) we get,
$\begin{align}
  & \text{Speed}=\dfrac{\text{Distance}}{\text{Time}} \\
 & \Rightarrow \dfrac{40}{3}=\dfrac{576+t}{90} \\
\end{align}$
The denominators of the above equation are divisible by 3 so dividing 3 by 3 we get 1 and dividing 90 by 3 we get 30.
$\dfrac{40}{1}=\dfrac{576+t}{30}$
On cross multiplying the above equation we get,
$1200=576+t$
Subtracting 576 on both the sides we get,
$\begin{align}
  & 1200-576=t \\
 & \Rightarrow 624=t \\
\end{align}$
As we have assumed that length of the tunnel is “t” and we have calculated the value of “t” as 624 m so the length of the tunnel is 624 m.

Note: This question demands the awareness and knowledge about changing the units of speed and time in such a way that when we put these values in the formula of speed, distance and time, the formula holds true.
If you forget to convert the units of speed, distance and time in such a way so that L.H.S is equal to R.H.S of the below formula then the value of “t” that you are getting is incorrect.
$\text{Speed}=\dfrac{\text{Distance}}{\text{Time}}$
And a mistake that you could do is by not considering the length of the train in the total distance traversed by the train. This mistake is quite possible because you might think that okay the train has traversed the length of the tunnel but here you are missing that the question is saying that the time taken by a train to cross the tunnel completely means the time from entering into the tunnel till the last part of the train exits through the tunnel and when the last part of the train exits from the tunnel then the train has traversed its length so the total length is the addition of tunnel length and the length of the train.