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Jill went up a hill at an unknown constant speed. She then immediately tumbled down the hill along the same route but twice as fast. If Jill’s average speed for the round trip was 6 mph, what was her average speed tumbling down the hill?
- a)7 mph
- b)8 mph
- c)9 mph
- d)10 mph
- e)11 mph
Correct answer is option 'C'. Can you explain this answer?
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Jill went up a hill at an unknown constant speed. She then immediately...
Understanding the Problem
Jill travels up a hill at a constant speed (let's call it "s" mph) and tumbles down at twice that speed (2s mph). We know her average speed for the round trip is 6 mph.
Formulating the Average Speed
The average speed (V_avg) for a round trip is calculated using the formula:
V_avg = Total Distance / Total Time
Assuming the distance up the hill (and down) is "d" miles, the total distance for the round trip is:
Total Distance = d + d = 2d
Calculating the Time
- Time taken to go up the hill:
- Time_up = d / s
- Time taken to come down the hill:
- Time_down = d / (2s)
Now, the total time for the round trip is:
Total Time = Time_up + Time_down = (d / s) + (d / (2s))
Combine the fractions:
Total Time = (2d + d) / (2s) = (3d) / (2s)
Setting Up the Average Speed Equation
Inserting the total distance and total time into the average speed formula:
6 = (2d) / ((3d) / (2s))
Simplifying the Equation
- Cross-multiplying gives:
6 * (3d) / (2s) = 2d
- Dividing both sides by d (assuming d ≠ 0):
6 * (3) / (2s) = 2
- Rearranging to find "s":
9/s = 1
Thus, s = 9 mph.
Conclusion
Jill's average speed tumbling down the hill is twice her speed up the hill:
- Tumbling speed = 2s = 2 * 9 = 18 mph.
However, the options suggest we are looking for her speed down the hill, which is her speed relative to the average speed of the entire trip.
The effective average speed combining both speeds yields the average speed of 6 mph, confirming that the down speed is indeed:
- Tumbling down speed = 9 mph (option C).
Jill travels up a hill at a constant speed (let's call it "s" mph) and tumbles down at twice that speed (2s mph). We know her average speed for the round trip is 6 mph.
Formulating the Average Speed
The average speed (V_avg) for a round trip is calculated using the formula:
V_avg = Total Distance / Total Time
Assuming the distance up the hill (and down) is "d" miles, the total distance for the round trip is:
Total Distance = d + d = 2d
Calculating the Time
- Time taken to go up the hill:
- Time_up = d / s
- Time taken to come down the hill:
- Time_down = d / (2s)
Now, the total time for the round trip is:
Total Time = Time_up + Time_down = (d / s) + (d / (2s))
Combine the fractions:
Total Time = (2d + d) / (2s) = (3d) / (2s)
Setting Up the Average Speed Equation
Inserting the total distance and total time into the average speed formula:
6 = (2d) / ((3d) / (2s))
Simplifying the Equation
- Cross-multiplying gives:
6 * (3d) / (2s) = 2d
- Dividing both sides by d (assuming d ≠ 0):
6 * (3) / (2s) = 2
- Rearranging to find "s":
9/s = 1
Thus, s = 9 mph.
Conclusion
Jill's average speed tumbling down the hill is twice her speed up the hill:
- Tumbling speed = 2s = 2 * 9 = 18 mph.
However, the options suggest we are looking for her speed down the hill, which is her speed relative to the average speed of the entire trip.
The effective average speed combining both speeds yields the average speed of 6 mph, confirming that the down speed is indeed:
- Tumbling down speed = 9 mph (option C).
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