Question

Meredith jogged to the top of a steep hill at an average pace of 6 miles per hour. She took the same trail back down. To her relief, the descent was much faster; her average speed rose to 14 miles per hour. If the entire run took Meredith exactly one hour to complete and she did not make any stops, how many miles, approximately, is the trail one way?

• A. 2
• B. 3
• C. 4
• D. 5
• E. 6

Hint:

For round-trip problems with the same distance but different speeds, setting up an equation based on total time is a very effective strategy. The formula \(T_{total} = \frac{d}{s_1} + \frac{d}{s_2}\) is fundamental here.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem involves two trips along the same path at different speeds, with a given total time. We can set up an equation relating the times for each part of the journey to the total time.
Step 2: Key Formula or Approach:
Let \(d\) be the one-way distance of the trail.
Let \(v_{up}\) be the speed going up and \(v_{down}\) be the speed going down.
Let \(t_{up}\) be the time going up and \(t_{down}\) be the time going down.
We know that Time = Distance / Speed. So, \(t_{up} = \frac{d}{v_{up}}\) and \(t_{down} = \frac{d}{v_{down}}\).
The total time is given: \(T_{total} = t_{up} + t_{down}\).
Step 3: Detailed Explanation:
From the problem, we have:
- Speed up, \(v_{up} = 6\) mph.
- Speed down, \(v_{down} = 14\) mph.
- Total time, \(T_{total} = 1\) hour.
Using the formulas from Step 2, we can set up the equation for the total time:
\[ \frac{d}{v_{up}} + \frac{d}{v_{down}} = T_{total} \] Substitute the known values into the equation:
\[ \frac{d}{6} + \frac{d}{14} = 1 \] To solve for \(d\), we need to find a common denominator for 6 and 14. The least common multiple (LCM) of 6 and 14 is 42.
Multiply the entire equation by 42 to eliminate the fractions:
\[ 42 \left( \frac{d}{6} \right) + 42 \left( \frac{d}{14} \right) = 42(1) \] \[ 7d + 3d = 42 \] Combine the terms with \(d\):
\[ 10d = 42 \] Solve for \(d\):
\[ d = \frac{42}{10} = 4.2 \text{ miles} \] The question asks for the approximate number of miles for the one-way trail. The value 4.2 is closest to 4.
Step 4: Final Answer:
The trail is approximately 4 miles one way.