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### Train and Tunnel

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Date: 7/18/96 at 7:32:4
From: Anonymous
Subject: Train, Tunnel Length

A train passes completely through a tunnel in 5 minutes. A second
train, twice as long, passes through the tunnel in 6 minutes. If both
trains are traveling at the same speed, 24km/h, determine the length
of the tunnel and the lengths of the trains.
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```
Date: 7/22/96 at 21:58:24
From: Doctor Luis
Subject: Re: Train, Tunnel Length

To answer this question you must first define what is meant by a
train "passing through a tunnel." Thus, in order to solve this, I will
assume that the time it takes for the train to pass through the tunnel
is the time interval measured from the instant the front of the train
enters the tunnel to the instant the back of the train leaves the
tunnel.

What is the distance traveled by the front while the train goes
"through" the tunnel?  Answer: T+L, where L is the length of the
train, and T is the length of the tunnel (this is because when the
back leaves the tunnel, the front is L meters away from the back).

So, the first train (length l) passes the tunnel (length T) in 5
minutes, at 24 km/hr. The distance covered (by the front) is

(24 km/hr)*(5 mins)*(1/60 hr/mins)*(1000 m/km) = 2000 m

Therefore, T + l = 2000 m

The second train (length 2*l) passes the tunnel (length T) in 6
minutes, at 24 km/hr. The distance covered (by the front) is

(24 km/hr)*(6 mins)*(1/60 hr/mins)*(1000 m/km) = 2400 m

Therefore, T + 2l = 2400 m

Now, we have two equations in two unknowns,

T + l = 2000
T +2l = 2400

from which the solution follows ( I get:  T = 1600 m, l = 400 m )

-Doctor Luis,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
Middle School Algebra
Middle School Word Problems

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