In 1km races ,runner 1 on track 1(with time 2 min,27.95 sec) appears to be faster than runner 2 on track 2(2min,28.15 sec). However ,length L2 of track! 2 might be slightly greater than L1 of track 1. How large can L2-L1 be for us still to conclude that runner 1 is faster?dl

To determine how much longer track 2 (L2) can be compared to track 1 (L1) while still concluding that runner 1 is faster, we need to analyze the times and speeds of both runners. Let's break it down step by step.

Understanding the Basics of Speed

Speed is defined as the distance traveled divided by the time taken. In this case, both runners are competing in a 1 km race, which is equivalent to 1000 meters. However, since the lengths of the tracks may differ, we need to account for that in our calculations.

Calculating the Speeds

First, let's convert the times taken by each runner into seconds for easier calculations:

• Runner 1: 2 minutes and 27.95 seconds = 147.95 seconds
• Runner 2: 2 minutes and 28.15 seconds = 148.15 seconds

Next, we can calculate their speeds:

• Speed of Runner 1 = Distance / Time = 1000 m / 147.95 s ≈ 6.75 m/s
• Speed of Runner 2 = Distance / Time = 1000 m / 148.15 s ≈ 6.75 m/s

Considering the Track Length Difference

Let’s denote the difference in track lengths as D = L2 - L1. If we want to find the maximum value of D such that Runner 1 is still considered faster, we can set up the following inequality:

Runner 1's time must be less than Runner 2's time adjusted for the longer distance:

147.95 < (148.15 + D / Speed of Runner 2)

Rearranging the Inequality

Now, let's rearrange this inequality to find D:

• 147.95 - 148.15 < D / Speed of Runner 2
• -0.20 < D / 6.75
• D > -0.20 * 6.75
• D > -1.35 meters

This means that for Runner 1 to still be considered faster, the length of track 2 can be at most 1.35 meters longer than track 1. If L2 is longer than L1 by more than this amount, we cannot conclude that Runner 1 is faster based solely on their times.

Final Thoughts

In summary, while Runner 1 appears to be faster based on their recorded times, the actual track lengths play a crucial role in determining who is truly faster. As long as the difference in track lengths does not exceed 1.35 meters, we can confidently say that Runner 1 is indeed the faster runner. This example illustrates the importance of considering all variables in a race, not just the times recorded.