Problem 25 - Precalculus Student Solutions Manual 5th

Question

Track Event Speeds At the 2008 Summer Olympics in Beijing. China, Usain Bolt (Jamaica) 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, 59 sec, also set in \(2008 ?\) (Hint: 1 m es 3.281 ft.) (Source: World Almanac and Book of Facts.)

Step-by-Step Solution

Answer

1 hr, 7 min, 35.82 sec. This is faster than the fastest marathon time of 2 hr, 3 min, 59 sec.

Step 1 - Determine Bolt's Speed

First, calculate Usain Bolt's speed in meters per second (m/s). His time for the 100-meter dash is 9.69 seconds. Use the formula for speed: \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \] So, Bolt's speed is: \[ \text{Speed} = \frac{100 \text{ m}}{9.69 \text{ s}} \text{ m/s}\] \[ \text{Speed} \text{ (in m/s)} = 10.32 \text{ m/s} \]

Step 2 - Convert Marathon Distance to Meters

Convert the marathon distance from miles to meters. We know that 1 mile = 1609.344 meters. The marathon distance is 26 miles. Thus, \[ \text{Distance (in m)} = 26 \text{ mi} \times 1609.344 \text{ m/mi} \] \[ \text{Distance (in m)} = 41842.944 \text{ m} \]

Step 3 - Calculate Time for the Marathon at Bolt's Speed

Using Bolt's speed from Step 1, calculate the time it would take to run the marathon distance. Use the formula for time: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] So, the time is: \[ \text{Time} = \frac{41842.944 \text{ m}}{10.32 \text{ m/s}} \] \[ \text{Time (in s)} = 4055.82 \text{ s} \] Convert this time from seconds to hours, minutes, and seconds. \[ 4055.82 \text{ s} = 1 \text{ hr, 7 min, 35.82 sec} \]

Step 4 - Compare with the Fastest Marathon Time

The fastest marathon time is 2 hours, 3 minutes, and 59 seconds. Convert this time to seconds for easy comparison: \[ \text{2 hr, 3 min, 59 sec} \] \[ = 2 \times 3600 \text{ s} + 3 \times 60 \text{ s} + 59 \text{ s} \] \[ = 7439 \text{ s} \] Compare 4055.82 seconds (Bolt's marathon time) with 7439 seconds (the fastest marathon time). Bolt's theoretical marathon time is significantly faster.

Key Concepts

Speed Calculation

Speed is a measure of how fast something moves over a specific distance in a certain amount of time. Understanding speed calculations is essential in problem-solving scenarios like the one involving Usain Bolt's 100-meter dash.
To find the speed, use the formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
In Bolt's case: \[ \text{Speed} = \frac{100 \text{ meters}}{9.69 \text{ seconds}} \approx 10.32 \text{ m/s} \] which means Bolt ran at a speed of 10.32 meters per second.

• This calculation helps us understand the pace at which Bolt moves.
• It's important to get the units right: Distance in meters and Time in seconds.
• Maintaining consistent units ensures the accuracy of your speed calculation.

This foundational concept is the first step in determining how quickly Bolt could potentially complete a marathon if he maintained that same speed.

Unit Conversion

Unit conversion involves changing a quantity expressed in one type of unit into another. This process is crucial when comparing different types of measurements. For example, to analyze the distance in Bolt's marathon, we need to convert miles to meters.
Here’s the marathon distance conversion step-by-step:

• We know that 1 mile equals 1609.344 meters.
• The marathon distance is 26 miles.
• To convert miles to meters, multiply the number of miles by the conversion factor: \[ 26 \text{ miles} \times 1609.344 \frac{\text{meters}}{\text{mile}} = 41842.944 \text{ meters} \]

By converting the marathon distance into meters, we align the units with Bolt's speed, calculated in meters per second. This conversion ensures consistent units, making subsequent time calculations accurate.

Time Comparison

Once we have Bolt’s marathon time, the next step is to compare it with the existing marathon record. This section focuses on converting and comparing time units efficiently.
First, we calculate Bolt's marathon time in seconds: \[ \text{Time} = \frac{41842.944 \text{ meters}}{10.32 \text{ meters/second}} \approx 4055.82 \text{ seconds} \] However, time comparison is easier with standard units (hours, minutes, seconds), so we convert seconds into these units:

• 1 hour = 3600 seconds
• 4055.82 seconds \[ = 1 \text{ hour} + \frac{4055.82 - 3600}{60} \approx 1 \text{ hour, 7 minutes, and 35.82 seconds} \]

Compare this with the fastest marathon record of 2 hours, 3 minutes, and 59 seconds, converted to seconds: \[ 2 \text{ hours} \times 3600 \text{ seconds/hour} + 3 \text{ minutes} \times 60 \text{ seconds/minute} + 59 \text{ seconds} = 7439 \text{ seconds} \] Clearly, Bolt’s theoretical marathon time of 4055.82 seconds is significantly faster than 7439 seconds. This comparison highlights the importance of consistent units and accurate conversion for a meaningful analysis.