Suppose a train traveling 90 mph takes four seconds to enter a tunnel. From there it takes one minute before the train is completely outside the tunnel. Find the length of both the train and the tunnel.
t = train length ... T = tunnel length
t = 90 mi/h * 5280 ft/mi * 4 s / 3600 s/hr
the train travels its length in 4 s
... it takes one minute (60 s) for the front of the train to traverse the tunnel
T = t (60 s / 4 s)
To solve this problem, we need to first determine the speed and distance traveled by the train in both the tunnel and outside the tunnel.
Let's start by converting the train's speed from miles per hour (mph) to feet per second (ft/s). Since 1 mile = 5280 feet, we have:
Speed of train = 90 mph = 90 * 5280 ft / 3600 s ≈ 132 ft/s
Now, let's consider the time it takes for the train to enter the tunnel. We are given that it takes 4 seconds. Since distance = speed × time, we can calculate the distance traveled by the train in those 4 seconds:
Distance = Speed × Time = 132 ft/s * 4 s = 528 ft
This distance is equal to the combined length of the train and the tunnel.
Next, we are given that it takes one minute (60 seconds) for the train to completely leave the tunnel. During this time, the train will travel an additional distance equal to its own length.
Let's represent the length of the train as "T" and the length of the tunnel as "L". To find their values, we can set up the following equation:
Distance = Train Length (T) + Tunnel Length (L) = 528 ft + T (distance outside the tunnel)
Since it takes one minute for the train to completely leave the tunnel, the distance traveled outside the tunnel is given by:
Distance outside the tunnel = Speed × Time = 132 ft/s * 60 s = 7920 ft
Plugging this into the equation, we have:
528 ft + T = 7920 ft
Subtracting 528 ft from both sides:
T = 7920 ft - 528 ft = 7392 ft
Thus, the length of the train is 7392 feet.
Finally, we can find the length of the tunnel by subtracting the length of the train from the combined distance of the train and the tunnel:
L = Distance - T = 528 ft - 7392 ft ≈ -6864 ft
But wait! The negative value for the length of the tunnel doesn't make sense in this context. It suggests that there might be an error or oversight in the given information or calculations.
Please double-check the provided data or review the problem to ensure its accuracy.