Problem 81 - College Algebra and Calculus: An Applied Approach

Question

New York City Marathon In Exercises 81 and 82, the length of the New York City Marathon course is 26 miles, 385 yards. Find the average speed of the record holding runner. (Note that 1 mile \(=5280\) feet \(=\) 1760 yards.) Men's record time: 2 hours, \(7 \frac{3}{4}\) minutes

Step-by-Step Solution

Answer

The average speed of the record holding runner is approximately 12.31 miles per hour.

Step 1: Converting all units to miles

The distance of the marathon course is given as 26 miles, 385 yards. We know that 1 mile is equal to 1760 yards. Hence, to convert 385 yards into miles, we divide 385 by 1760. That gives us \( \frac{385}{1760} \approx 0.218\) miles.

Step 2: Calculating total distance in miles

The total distance of the marathon is the sum of miles and the converted distance from yards to miles. So, total distance = 26 miles + 0.218 miles = 26.218 miles.

Step 3: Converting time to hours

The Men's record time given is 2 hours, \(7 \frac{3}{4}\) minutes. We need to convert this time into hours for a consistent unit with our mile measurement. We know that 1 hour = 60 minutes, so we divide \(7 \frac{3}{4}\) by 60 to convert it into hours, which gives approximately 0.129 hour.

Step 4: Calculating total time in hours

The total time taken is the sum of hours and the time converted from minutes to hours. So, total time = 2 hours + 0.129 hour = 2.129 hours.

Step 5: Formulating speed equation and calculate speed

Average speed is calculated by dividing the total distance by the total time. Therefore, average speed \( = \frac{26.218 miles}{2.129 hours} \approx 12.31\) miles per hour. We approximate to two decimal places.

Key Concepts

Unit Conversion

To solve problems involving different units of measurement, converting these units into a common unit is crucial. This allows for accurate calculations and comparisons. In the context of the New York City Marathon exercise, we needed to convert yards into miles to find the distance completed by the runner in one consistent unit.

• Identify the original units: The marathon distance is given as 26 miles and 385 yards.
• Understand conversion rates: We know 1 mile equals 1760 yards.
• Apply the conversion: By dividing 385 yards by 1760, we find the equivalent in miles, which is approximately 0.218 miles.

Combine the converted distance with the given miles, resulting in the total distance covered in miles. This step is essential in ensuring that all measurements align for further calculation.

Problem Solving

Problem-solving involves understanding what the question is asking and determining the steps required to reach a solution. Here, you are asked to find the average speed of a marathon runner. You must first interpret the problem by identifying key components: the distance and time.

• Read the problem carefully: Note the given data – total marathon distance and record time.
• Identify what is needed: You need an average speed, which depends on distance and time.
• Plan the steps: Break these into smaller tasks, such as unit conversion and average speed calculation.

By following this logical plan, you ensure that every piece of information is accounted for, leading you confidently to the desired answer.

Distance and Time

Distance and time are two fundamental concepts in motion problems like calculating average speed. Understanding how these two relate is key to finding solutions in velocity problems.
### Distance The marathon distance is a crucial part of the problem. After converting the additional 385 yards into miles, the total distance becomes 26.218 miles.
### Time Time is given in hours and minutes, necessitating conversion for consistency with distance units. The record time, 2 hours and 7.75 minutes, was converted into hours as 2.129 hours.
### Connecting Distance and Time Once both distance and time are in appropriate units, you can calculate average speed: dividing distance by time gives the rate in miles per hour. Thus, using these clear conversions and calculations, you bridge the raw data with the final desired outcome.

Algebraic Manipulation

Algebraic manipulation refers to the methods of rearranging formulas to solve for a desired variable. In the realm of average speed, simple algebra helps bridge the gap between known and unknown values. Average speed is mathematically expressed as:

• Speed = Distance / Time

For the marathon problem:

• Distance = 26.218 miles
• Time = 2.129 hours

Plug these into the formula to find:

• Average speed = 26.218 / 2.129
• This yields approximately 12.31 miles per hour.

Algebraic manipulation is the tool by which you can rearrange and substitute values into formulas, providing clarity and precision in solving such physics-related problems.