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Chapter 9: Problem 65

A train \(350 \mathrm{~m}\) long is running at the speed of \(36 \mathrm{~km} / \mathrm{h}\). If it crosses a tunnel in 1 minute, then the length of the tunnel (in metres) is: (a) \(200 \mathrm{~m}^{--}\) (b) \(.50 \mathrm{~m}^{-}\) (c) \(150 \mathrm{~m}\) (d) none of these

Short Answer

Expert verified
Answer: The length of the tunnel is 250 meters.

Step by step solution

01

Identify the given information

We are given the following information: - The length of the train: \(350 \mathrm{~m}\) - The speed of the train: \(36 \mathrm{~km} / \mathrm{h}\) - The time it takes to cross the tunnel: 1 minute
02

Convert the speed to meters per minute

We need to convert the speed of the train to \(\mathrm{m} / \mathrm{minute}\) to match the time unit mentioned in the problem. 1 km is equal to 1000 meters and 1 hour is equal to 60 minutes, so: \(36 \frac{\mathrm{km}}{\mathrm{h}} \cdot \frac{1000 \mathrm{m}}{1 \mathrm{km}} \cdot \frac{1 \mathrm{h}}{60 \mathrm{min}} = 600 \frac{\mathrm{m}}{\mathrm{min}}\) So, the speed of the train is \(600 \mathrm{~m} / \mathrm{min}\).
03

Calculate the total distance traveled by the train

Use the formula for distance: Distance = Speed 脳 Time In this case: Distance = \((600 \mathrm{~m} / \mathrm{min}) \cdot (1 \mathrm{~min}) = 600 \mathrm{~m}\)
04

Determine the length of the tunnel

The total distance traveled (600 m) includes the length of the train (350 m) and the length of the tunnel (x m). So we can set up the equation: Length of the tunnel (x) = Total distance - Length of the train x = 600 m - 350 m x = 250 m So, the length of the tunnel is 250 meters, which corresponds to the answer (b).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Conversion
To solve the problem of the train crossing the tunnel, we need to perform unit conversion to ensure that all measurements work together efficiently. The train's speed is given in kilometers per hour (km/h), but the time taken to cross the tunnel is in minutes. We need to convert the speed to meters per minute (m/min) to match the time unit.

Here's the straightforward conversion process:
  • First, convert the speed from kilometers to meters by multiplying by 1000 (since 1 km = 1000 meters).
  • Next, convert hours to minutes by dividing by 60 (since 1 hour = 60 minutes).
Thus, the train's speed converts from 36 km/h to 600 m/min, making it compatible with the minute-based time frame in the problem.
Equation Setup
Setting up the right equation is crucial to finding the solution. With the converted speed and given time, you can use the basic formula for distance:

Distance = Speed 脳 Time.

This formula tells us the total distance the train covers while crossing the tunnel. However, the total distance includes both the length of the train and the length of the tunnel. Therefore, the equation becomes:

Total distance = Length of train + Length of tunnel.

By rearranging this equation, we can solve for the length of the tunnel. This setup allows us to identify all known variables and determine the unknown variable as effectively as possible.
Problem Solving Steps
Problem solving is a process that often involves breaking down the problem into smaller parts. Follow these steps to untangle the puzzle:

1. **Identify** what is given and what needs to be found. We know the speed, time, and length of the train, and we need to find the length of the tunnel.
2. **Convert** units where necessary so all measurements align.
3. **Set Up** the equation using the basic distance formula and adapt it to the problem's context.
4. **Solve** the equation to find the unknown using simple arithmetic operations.
This systematic approach ensures clarity and simplifies solving any involved word problems.
Mathematical Calculation
Once all components are properly converted and the equation is set up, it's time for calculation.

With the formula for distance (Distance = Speed 脳 Time) set as Total distance = Length of train + Length of tunnel, we use the known quantities to solve for the unknown.

Total distance covered by the train is calculated as 600 m (since Speed = 600 m/min and Time = 1 min). Now apply the calculation:
  • Total distance (600 m) - Length of the train (350 m) = Length of the tunnel.
So, Length of the tunnel = 600 m - 350 m = 250 m.

Therefore, the tunnel's length is 250 meters, neatly solving the problem.

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Most popular questions from this chapter

A boat covers 48 km upstream and 72 km downstream in 12 hours, while it covers \(72 \mathrm{~km}\) upstream and \(48 \mathrm{~km}\) downstrenm in 13 hours. The speed of stream is: (a) \(2 \mathrm{~km} / \mathrm{h}\) (b) \(2.2 \mathrm{~km} / \mathrm{h}\) (c) \(2.5 \mathrm{~km} / \mathrm{h}\) (d) \(4 \mathrm{~km} / \mathrm{h}\)

The relative speed of minute-hand with respect to hour-hand is : (a) \(\left(5 \frac{1}{2}\right)^{\circ}\) per minute (b) \(\left(\frac{11}{12}\right)\) minute per minute (c) \(6^{\text {" per minute }}\) (d) both (a) and (b)

In a kilometre race, \(A\) can give B a start of \(20 \mathrm{~m}\) and also in a half kilometre race \(\mathrm{C}\) beats \(\mathrm{A}\) by \(50 \mathrm{~m} .\) \(B\) and \(C\) run a half \(\mathrm{km}\) race, who should give a start to the slower runner and of how many metres so that they both finish the race at the same time? (a) \(C, 59 \mathrm{~m}\) (b) B, \(34 \mathrm{~m}\) (c) C, \(48 \mathrm{~m}\) (d) \(B, 56 \mathrm{~m}\)

A train met with an accident \(120 \mathrm{~km}\) from station \(A\). It completed the remaining joumey at \(5 / 6\) of its previous speed and reached 2 hours lare at station \(B\). Had the accident taken place \(300 \mathrm{~km}\) further, it would have been only 1 hour late? Whar is the speed of the train? (a) \(100 \mathrm{~km} / \mathrm{h}\) (b) \(120 \mathrm{~km} / \mathrm{h}\) (c) \(60 \mathrm{~km} / \mathrm{h}\) (d) \(50 \mathrm{~km} / \mathrm{h}\)

Walking at four fifth of his usual speed Vijay Malya reaches his office 15 minutes late on a particular day. The next day, he walked at \(5 / 4\) of his usual speed. How early would he be to the office when compared to the previous day? (a) \(27 \mathrm{~min}\) (b) 32 min (c) \(30 \mathrm{~min}\) (d) none of these
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