Calculating the Time for Anders and Benj to Meet on a Circular Track: A Comprehensive Guide

Running in opposite directions on a circular track can be an intriguing mathematical problem. In this article, we will explore how to calculate the time until Anders and Benj, running from the same starting point but in opposite directions, meet. This problem involves understanding the circumference of the track, the concept of relative speed, and the conversion of units from kilometers per hour to meters per second. Let's dive in!

Understanding the Problem

Anders and Benj start from the same point on a circular track with a diameter of 800 meters. Anders runs at an average speed of 3 kilometers per hour (kph), while Benj runs at 4 kph. To find the time they will meet, we need to follow these steps:

Step 1: Calculate the Circumference of the Track

First, we need to determine the circumference of the track. The formula for the circumference ($C$) of a circle is:

[ C pi times d ]

Given that the diameter ($d$) of the track is 800 meters, we can calculate the circumference:

[ C pi times 800 approx 2513.27 text{ meters} ]

Step 2: Determine the Combined Speed

Since Anders and Benj are running in opposite directions, their relative speed is the sum of their individual speeds. Anders' speed is 3 kph, and Benj's speed is 4 kph. Their combined speed is:

[ 3 text{ kph} 4 text{ kph} 7 text{ kph} ]

Step 3: Convert Combined Speed to Meters per Second

To use the speed in the same units as the distance, we need to convert kilometers per hour to meters per second. The conversion factor is:

[ 1 text{ kph} frac{1000 text{ meters}}{3600 text{ seconds}} frac{5}{18} text{ m/s} ]

Thus, the combined speed in meters per second is:

[ 7 text{ kph} times frac{5}{18} approx 1.944 text{ m/s} ]

Step 4: Calculate the Time to Meet

The time ($t$) required for Anders and Benj to meet is the distance they need to cover, which is the circumference of the track, divided by their combined speed:

[ t frac{C}{text{Combined speed}} frac{2513.27 text{ meters}}{1.944 text{ m/s}} approx 1292.24 text{ seconds} ]

Step 5: Convert the Time to Minutes

To convert the time into minutes, we divide by 60:

[ t approx frac{1292.24}{60} approx 21.54 text{ minutes} ]

Alternative Solution

An alternative approach to this problem involves directly working with the given information without converting units. We can use the relationship $D R times t$ to solve for the time:

Step 1: Determine the Combined Distance Traveled

If the circumference of the track is 800$pi$ meters, the combined distance traveled until they meet is the circumference itself, which is 800$pi$ meters. The combined speed in meters per second is $frac{15}{7}$ m/s (since 3 kph $frac{5}{6}$ m/s and 4 kph $frac{10}{9}$ m/s, their combined speed is $frac{5}{6} frac{10}{9} frac{15}{7}$ m/s).

[ t frac{text{Distance}}{text{Combined speed}} frac{800pi text{ meters}}{frac{15}{7} text{ m/s}} frac{800pi times 7}{15} approx 1293 text{ seconds} ]

Step 2: Convert the Time to Minutes

Converting 1293 seconds to minutes:

[ t approx frac{1293}{60} approx 21.55 text{ minutes} ]

Conclusion

Therefore, Anders and Benj will meet after approximately 21.54 minutes. This time can be confirmed by the alternative solution, which gives 21.55 minutes. Both methods utilize the principles of relative speed and the relationship between distance, speed, and time.