Question

A competitive college runner ran a \(5-\mathrm{K}(5.0-\mathrm{km})\) race in \(15 \mathrm{~min}\), \(23 \mathrm{~s}\). What was her pace in miles per hour?

Short answer

Using the steps, the runner's pace is approximately 12.1 miles per hour.

Step-by-step solution

Convert Time to Hours

First, convert the time from minutes and seconds to hours. The runner's time is 15 minutes and 23 seconds. Convert minutes to hours by dividing by 60. Convert seconds to hours by dividing by 3600.

Convert Kilometers to Miles

Convert the race distance from kilometers to miles. Use the conversion factor: 1 kilometer = 0.621371 miles.

Calculate Pace

Divide the distance in miles by the time in hours to find the runner's pace in miles per hour.

Key concepts

Unit Conversion

Understanding unit conversion is crucial for solving problems in different scientific, mathematical, and real-life contexts. In this exercise, we need to convert time from minutes and seconds to hours and distance from kilometers to miles.

To convert time from minutes to hours, you divide by 60, since there are 60 minutes in an hour. For example, 15 minutes is converted by doing: \[ 15 \text{ minutes} \times \frac{1 \text{ hour}}{60 \text{ minutes}} = 0.25 \text{ hours} \]

Next, we convert seconds to hours. There are 3600 seconds in an hour, so we divide by 3600. Thus, 23 seconds is converted by: \[ 23 \text{ seconds} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} = 0.00639 \text{ hours} \]

This same method applies to any unit conversion. Remember to always multiply by a fraction where the unit you want to cancel out is in the denominator.

Speed Calculation

Speed calculation is often essential in many problems, including finding how fast something or someone is moving. Speed is defined as the distance covered over a specified amount of time. The formula for speed is: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]

In our specific problem, after converting the time run to hours and the distance run to miles, we use this formula to find the pace. For example, if the converted time is approximately 0.25639 hours and the converted distance is about 3.107 miles, the pace calculation would be: \[ \text{Pace} = \frac{3.107 \text{ miles}}{0.25639 \text{ hours}} \] This result gives the speed or pace of the runner in miles per hour. Understanding how to properly calculate speed is useful in analyzing motion and efficiency.

Metric to Imperial Units

Converting from metric to imperial units is a valuable skill, especially in countries like the United States where both systems are commonly used. In this exercise, we need to convert kilometers (metric unit) to miles (imperial unit). The conversion factor to remember is that 1 kilometer equals approximately 0.621371 miles.

To convert the 5-kilometer race to miles, you multiply: \[ 5 \text{ kilometers} \times 0.621371 \frac{\text{miles}}{\text{kilometers}} = 3.107 \text{ miles} \]

This methodology applies to any metric-to-imperial conversion. Always multiply by the appropriate conversion factor to get the result in your desired units. By mastering these conversions, you can easily understand and work within both systems of measurement.