Calculate Distance Using Speed and Time Difference
This problem involves finding the distance covered by two runners, A and B, given their speeds and the difference in the time they take to cover the same distance.
Identify Given Information
- Speed of Runner A: \(v_A = 15\) km/h
- Speed of Runner B: \(v_B = 16\) km/h
- Time difference: Runner A takes 16 minutes more than Runner B.
Convert Time Difference to Hours
The time difference is given in minutes, so we convert it to hours for consistency with the speed units (km/h).
\( \Delta t = 16 \text{ min} = \frac{16}{60} \text{ hours} \)Set Up the Equation
Let \(d\) be the distance covered (in km). The time taken by each runner is \(t = \frac{d}{v}\).
- Time taken by A: \(t_A = \frac{d}{v_A} = \frac{d}{15}\) hours
- Time taken by B: \(t_B = \frac{d}{v_B} = \frac{d}{16}\) hours
Since A takes more time than B, the difference is \(t_A - t_B = \Delta t\).
\( \frac{d}{15} - \frac{d}{16} = \frac{16}{60} \)Solve for Distance
Factor out the distance \(d\) and find a common denominator for the speeds.
\( d \left( \frac{1}{15} - \frac{1}{16} \right) = \frac{16}{60} \)Calculate the term in the parenthesis:
\( \frac{1}{15} - \frac{1}{16} = \frac{16 - 15}{15 \times 16} = \frac{1}{240} \)Substitute this back into the equation:
\( d \left( \frac{1}{240} \right) = \frac{16}{60} \)Now, solve for \(d\):
\( d = \frac{16}{60} \times 240 \) \( d = 16 \times \frac{240}{60} \) \( d = 16 \times 4 \) \( d = 64 \)Therefore, the distance is 64 km.
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