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Question

Difficulty: MediumRate, Time, and Distance Problems

A cyclist travels along a scenic mountain road from a base lodge to a summit point at a constant speed of 1212 miles per hour. The cyclist then returns along the exact same route from the summit point back to the base lodge at a constant speed of 2424 miles per hour. If the entire round-trip ride takes a total of 33 hours, what is the total distance, in miles, for the entire round trip?

Answer: 48 miles

Answer

The total distance for the entire round trip is 48 miles.
To find the total round-trip distance, let dd be the one-way distance. The time spent going uphill is d12\frac{d}{12} hours and downhill is d24\frac{d}{24} hours. Combining these gives d12+d24=3\frac{d}{12} + \frac{d}{24} = 3, which simplifies to 3d24=3\frac{3d}{24} = 3 or d=24d = 24 miles. Since the trip covers both legs, the total distance is 2d=482d = 48 miles.

Step-by-Step Solution

1
Define the variable for distance.
Let dd represent the one-way distance in miles between the base lodge and the summit point.
Establishing a variable for the unknown leg distance allows setting up rate-time-distance relationships.
2
Express time in terms of distance and rate for both legs.
Uphill time t1=d12t_1 = \frac{d}{12} hours; Downhill time t2=d24t_2 = \frac{d}{24} hours.
Using Time=DistanceRate\text{Time} = \frac{\text{Distance}}{\text{Rate}} converts given speeds into expressions for duration.
3
Formulate and solve the total time equation.
\frac{d}{12} + \frac{d}{24} = 3 \implies \frac{3d}{24} = 3 \implies d = 24 \text{ miles}.
The sum of the time taken for both legs equals the given total round-trip time of 3 hours.
4
Compute the total round-trip distance.
\text{Total distance} = 2 \times d = 2 \times 24 = 48 \text{ miles}.
The question specifically asks for the total distance of the entire round trip, which consists of two equal legs of length dd.

Key Concept

Harmonic Mean and Round-Trip Rate Problems
Estimated Time:2m 0s
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Topics

Rate, Time, and Distance Problems

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