A train is travelling at a constant speed of 75 miles per hour and enters a tunnel that is 3.5 miles long. The train itself is 0.25 mile long. How long, in minutes, does it take from the moment the front of the train enters the tunnel until the moment the rear of the train emerges completely from the tunnel?

Introduction / Context:
This tunnel problem tests understanding of how much distance a train must cover to clear a tunnel completely. It also checks correct handling of miles and hours units and converting the final answer into minutes. Such questions are common in exams because they combine geometry (lengths) with uniform motion concepts.

Given Data / Assumptions:

• Speed of the train = 75 miles per hour.
• Length of the tunnel = 3.5 miles.
• Length of the train = 0.25 mile.
• The train moves at a constant speed with no acceleration or deceleration inside the tunnel.
• We need the total time from front entering until rear leaving, expressed in minutes.

Concept / Approach:
To clear the tunnel completely, the rear end of the train must move past the far end of the tunnel. Therefore, the total distance traveled by the train's front from entry to final exit is the sum of the tunnel length and the train length. Once we know this effective distance and the constant speed, we can find the time using the basic formula time = distance / speed. Finally, we convert the time from hours to minutes by multiplying by 60.

Step-by-Step Solution:
Step 1: Effective distance to be covered = tunnel length + train length.Step 2: Tunnel length = 3.5 miles, train length = 0.25 mile, so total distance = 3.5 + 0.25 = 3.75 miles.Step 3: Speed of the train = 75 miles per hour.Step 4: Time in hours = distance / speed = 3.75 / 75.Step 5: Simplify 3.75 / 75 = 0.05 hours.Step 6: Convert hours to minutes: time in minutes = 0.05 * 60 = 3 minutes.

Verification / Alternative check:
We can think of 75 miles per hour as 1.25 miles per minute, since 75 / 60 = 1.25.If the train needs to cover 3.75 miles, time in minutes = 3.75 / 1.25 = 3 minutes.This alternative view confirms that the previous calculation is correct.

Why Other Options Are Wrong:
Values like 2.5 minutes or 3.2 minutes come from rounding too early or using incorrect effective distance. The option 3.5 minutes could result from erroneously adding extra distance or misinterpreting the required time. The value 4 minutes is clearly larger and would imply a lower speed than given in the question. Only 3 minutes satisfies the precise distance and speed relationship.

Common Pitfalls:
Candidates sometimes forget to add the train length to the tunnel length and mistakenly use only the tunnel's length. Others confuse miles per hour with miles per minute and do not convert correctly to minutes. It is also easy to mix up when to multiply or divide by 60 during unit conversion. Keeping track of units at every step helps prevent these errors.

Final Answer:
The train takes 3 minutes to completely pass through the tunnel.