Problem 48 - Thomas Calculus

Question

A marathoner ran the 26.2-mi New York City Marathon in 2.2 hours. Show that at least twice the marathoner was running at exactly 11 mph.

Step-by-Step Solution

Answer

The marathoner ran at exactly 11 mph at least twice, by the Intermediate Value Theorem.

Step 1: Identify Key Information

For this problem, we know the marathon is 26.2 miles long and the marathoner completed it in 2.2 hours. We are tasked with proving that the marathoner ran at least twice at an exact speed of 11 mph.

Step 2: Calculate Average Speed

Calculate the average speed by dividing the total distance (26.2 miles) by the total time (2.2 hours): \[ \text{Average speed} = \frac{26.2 \text{ miles}}{2.2 \text{ hours}} = 11.909 \text{ mph} \]

Step 3: Apply the Intermediate Value Theorem

The Intermediate Value Theorem (IVT) states if a function is continuous on a closed interval \(a, b\), then it takes every value between \(f(a)\) and \(f(b)\). The marathoner's speed is a continuous function over the marathon, so by IVT, they must have reached 11 mph at least twice to average 11.909 mph over 2.2 hours.

Step 4: Conclusion

Since the average speed is greater than 11 mph and the speed function is continuous, by the Intermediate Value Theorem, the speed would have been exactly 11 mph at least twice during the marathon.

Key Concepts

Continuous Functions

In mathematics, a continuous function is a type of function where small changes in the input produce small changes in the output. Think of it as drawing a line on paper without lifting your pencil. This type of function is crucial when analyzing real-world scenarios like a marathon runner's speed.

• Continuous functions allow us to use powerful mathematical theorems like the Intermediate Value Theorem (IVT).
• In the context of a marathon, the runner's speed from start to finish can be seen as a continuous function of time.

Since there are no abrupt jumps or breaks in how speed changes, if the runner changes speed smoothly, the function modeling their speed over time will be continuous.

Average Speed

Average speed is a simple but important concept. It helps us understand overall performance by giving us a single value representing the total distance traveled over a given time period. The formula to calculate average speed is:\[\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\]

• In the exercise, the marathoner's total distance is 26.2 miles, and the total time is 2.2 hours.
• This results in an average speed of 11.909 mph, slightly above 11 mph.

Understanding average speed is essential in linking the distance, time, and the application of the Intermediate Value Theorem to find specific moments, like when the speed was exactly 11 mph.

Marathon Distance

The New York City Marathon spans a total distance of 26.2 miles. Marathons are standardized races, which means every runner competes over this same distance. This consistency in measurement makes it straightforward to calculate speeds and timings throughout the race.

• The standard length helps us apply mathematical reasoning in problems involving average and instantaneous speed.
• Knowing the marathon length allows us to compute precise averages, which lead us to deeper insights such as occurrences of specific speeds using continuous functions.

This information is essential in solving problems like the one presented, where knowing the distance allows us to leverage mathematical theorems effectively.

Proof by Theorem

Proof by theorem is a method used to demonstrate mathematical truths using established theorems. In this exercise, the Intermediate Value Theorem (IVT) serves as the core mathematical principle leveraged to prove the runner's speed reached an exact value twice.

Using the Intermediate Value Theorem

• The IVT states that if a continuous function takes two values at two points, it must take every value in between at least once.
• For our marathon runner, the speed function is continuous, and the overall average speed is 11.909 mph.
• To reach an average of 11.909 mph, the runner's speed must have fallen back to exactly 11 mph at least twice within the continuous interval of 0 to 2.2 hours.

Applying proof by theorem helps confirm our conclusion and provides a logical framework to understand when and how specific values must occur in continuous functions.