Solve each problem. See Example 2. At the 2008 Summer Olympics in Beijing, Usain Bolt set a new Olympic and world record in the 100-m dash with a time of 9.69 sec. If this pace could be maintained for an entire 26-mi marathon, what would his time be? How would this time compare to the fastest time for a marathon, which is 2 hr, 3 min, 23 sec, set in 2013? (Hint: 1 m ≈ 3.281 ft.) (Data from Sports Illustrated Almanac.)

Verified step by step guidance

1. First, convert the marathon distance from miles to feet. Since 1 mile = 5280 feet, multiply 26 miles by 5280 feet/mile to get the total distance in feet: \(26 \times 5280\) feet.
2. Next, find Usain Bolt's speed in feet per second. Use the given 100-meter dash time of 9.69 seconds and convert 100 meters to feet using the conversion \(1 \text{ m} \approx 3.281 \text{ ft}\). Calculate speed as \(\text{speed} = \frac{\text{distance}}{\text{time}} = \frac{100 \times 3.281}{9.69}\) feet per second.
3. Then, calculate the total time it would take Bolt to run the marathon distance at this speed by dividing the marathon distance in feet by his speed in feet per second: \(\text{time} = \frac{\text{marathon distance in feet}}{\text{speed in feet/second}}\) seconds.
4. Convert this total time from seconds into hours, minutes, and seconds for easier comparison. Use the fact that 1 hour = 3600 seconds and 1 minute = 60 seconds.
5. Finally, compare this calculated marathon time to the fastest marathon time given (2 hours, 3 minutes, 23 seconds) by subtracting one from the other or by discussing which is faster and by how much.

Key Concepts

Unit Conversion

Unit conversion involves changing measurements from one unit to another, such as meters to feet or miles to feet. This is essential for comparing distances or speeds given in different units. Accurate conversion ensures consistency in calculations, like converting marathon miles to meters or feet to match the pace units.

Rate and Proportion

Rate and proportion relate to comparing quantities with different units, such as speed (distance per time). Understanding how to scale a known rate (Usain Bolt's pace) to a different distance (marathon length) allows calculation of total time by setting up proportional relationships.

Time Calculation and Comparison

Time calculation involves converting units of time (seconds, minutes, hours) and summing or comparing durations. Comparing the calculated marathon time to the existing record requires converting all times to a common format to determine which is faster and by how much.