Question

The men's world record for running a mile outdoors (as of 1999 ) is 3 min 43.13 s. At this rate, how long would it take to run a \(2 \mathrm{~km}\) race \((1 \mathrm{mi}=1609 \mathrm{~m})\) ?

Short answer

4 minutes and 37.53 seconds

Step-by-step solution

Convert the World Record Time to Seconds

Firstly, we convert the record time of 3 minutes and 43.13 seconds into total seconds. We do this because calculations will be easier in one unit.\[ 3 \text{ min} = 3 \times 60 = 180 \text{ seconds} \]Adding 43.13 seconds to this, we get:\[ 180 + 43.13 = 223.13 \text{ seconds} \]

Calculate Speed in Meters per Second

Next, we calculate the speed in meters per second using the world record time. The distance for a mile is 1609 meters.\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{1609 \, \text{m}}{223.13 \, \text{s}} \approx 7.21 \, \text{m/s} \]

Calculate Time for a 2 km (2000 m) Race

Now, using the speed calculated in Step 2, we determine the time it would take to run 2000 meters.\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{2000 \, \text{m}}{7.21 \, \text{m/s}} \approx 277.53 \, \text{seconds} \]

Convert Total Time Back to Minutes and Seconds

Finally, we convert the time from seconds back to minutes and seconds for better understanding.\[ 277.53 \, \text{seconds} = 4 \, \text{minutes} \, 37.53 \, \text{seconds} \]Thus, it would take approximately 4 minutes and 37.53 seconds to run 2 km at the world record pace.

Key concepts

Speed Calculation

Speed is the measure of how fast something moves over a specific distance. To calculate speed, you use the formula:

• Speed = Distance / Time

In the original exercise, the speed of the runner is calculated after converting the running time for a mile into seconds. The runner covers a distance of 1609 meters in approximately 223.13 seconds, hence the speed is found to be approximately 7.21 meters per second.
Understanding speed calculation is crucial to solving problems where distance and time are involved, as it helps in determining how long it will take to cover a new distance given a constant speed.

Distance Conversion

Distance conversion involves changing the units of distance to align with units of time, so that calculations can become more cohesive. In this context, converting between miles and kilometers or meters is fundamental.

• 1 mile is equal to 1609 meters.

The original exercise specifically provides the length of a mile in meters. This allows for a seamless transition into calculating speed in meters per second rather than miles per minute or hour, making further calculations more straightforward since speed and time units are aligned.

Time Calculation

Time calculation is about determining the amount of time needed to cover a distance at a known speed. The formula for this calculation is:

• Time = Distance ÷ Speed

In the exercise, after finding the speed in meters per second, the time it takes to complete a 2-kilometer race is calculated using this formula. With a speed of 7.21 meters per second, it's calculated that a 2000-meter distance is covered in approximately 277.53 seconds.
This highlights the importance of accurately converting and calculating time to align with given units of speed and distance.

Unit Conversion

Unit conversion is the process of converting a given unit into a different unit without changing the actual quantity involved. This exercise requires converting minutes to seconds and meters to kilometers, ensuring uniformity in calculations:

• To convert minutes to seconds, multiply the number of minutes by 60.
• For distances, remember the equivalence: 1 km = 1000 m.

By using these basic conversion principles, the exercise becomes much easier to solve, as every calculation step remains consistent in its use of units. Proper understanding and application of unit conversions help avoid common calculation mistakes in multiple-step problems.