To analyze the race results between Alex and Ben, we can set up the information systematically.
- First Race:
- Distance of the race: 200 yards.
- When Alex finishes the race (200 yards), Ben is 15 yards behind. Therefore, when Alex has run 200 yards, Ben has only run \( 200 - 15 = 185 \) yards.
Let's denote:
- \( t_A \) = time it took Alex to run 200 yards
- \( t_B \) = time it took Ben to run 185 yards
Since both run the same time duration in the first race:
- Alex's speed \( v_A = \frac{200}{t_A} \)
- Ben's speed \( v_B = \frac{185}{t_B} \)
Since they ran at the same time, \[ t_A = t_B \]
This gives us the equations: \[ v_A \cdot t_A = 200 \quad \text{and} \quad v_B \cdot t_B = 185 \]
From the above, we can express \( v_B \) in terms of \( v_A \): \[ v_A = \frac{200}{t_A} \] \[ v_B = \frac{185}{t_A} \] Thus, \[ \frac{v_B}{v_A} = \frac{185}{200} = \frac{37}{40} \]
This implies that Ben runs at 37/40 of the speed of Alex.
- Second Race:
- In the second race, Alex starts 15 yards behind, meaning he runs 200 + 15 = 215 yards.
- Ben runs the usual 200 yards.
We can determine how long it takes each to finish the second race:
Let \( t_{A2} \) be the time for Alex in the second race: \[ t_{A2} = \frac{215}{v_A} \]
And Ben's time \( t_{B2} \) is: \[ t_{B2} = \frac{200}{v_B} \]
Since we established \( v_B = \frac{37}{40} v_A \), we can rewrite Ben's time: \[ t_{B2} = \frac{200}{\frac{37}{40} v_A} = \frac{200 \cdot 40}{37 v_A} = \frac{8000}{37 v_A} \]
Now we can compare both times:
- For Alex: \[ t_{A2} = \frac{215}{v_A} \]
- For Ben: \[ t_{B2} = \frac{8000}{37 v_A} \]
To compare: We compare \( t_{A2} \) to \( t_{B2} \). Setting up the comparison: \[ \frac{215}{v_A} \quad \text{vs} \quad \frac{8000}{37 v_A} \]
Since \( v_A \) is common in both expressions, we can ignore \( v_A \) when comparing: \[ 215 \quad \text{vs} \quad \frac{8000}{37} \]
Calculating \( \frac{8000}{37} \): \[ \frac{8000}{37} \approx 216.22 \]
Thus, \( 215 < 216.22 \) implies \( t_{A2} < t_{B2} \). This means Alex finishes before Ben in the second race.
- By how many yards did Alex win? To find out the number of yards Alex wins by, we calculate how far Ben runs while Alex finishes: Using the time for Alex \( t_{A2} = \frac{215}{v_A} \): \[ \text{Distance Ben covers in time } t_{A2} = v_B \cdot t_{A2} = \frac{37}{40}v_A \cdot \frac{215}{v_A} = \frac{37}{40} \cdot 215 = \frac{37 \times 215}{40} \] Calculating the distance. \[ \text{Distance Ben covered} = \frac{37 \times 215}{40} \approx 199.875 \text{ yards} \] Thus, the distance Ben fell short of the finish line: \[ 200 - 199.875 = 0.125 \text{ yards} \]
Therefore, Alex wins the second race by about 0.125 yards, or approximately 1.5 inches.