Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

In one version of Zeno's paradox, Achilles can run ten times as fast as the tortoise, but the tortoise has a 100 -yard headstart. Achilles cannot catch the tortoise, says Zeno, because when Achilles runs 100 yards the tortoise will have moved 10 yards ahead, when Achilles runs another 10 yards, the tortoise will have moved 1 yard ahead, and so on. Convince Zeno that Achilles will catch the tortoise and tell him exactly how many yards Achilles will have to run to do it.

Step-by-Step Solution

Answer

Achilles will catch the tortoise after running approximately 111.11 yards.

Step 1: Understanding the Problem

We need to determine at what distance Achilles overtakes the tortoise. Achilles runs ten times faster than the tortoise and starts 100 yards behind. Each time Achilles reaches the point where the tortoise was, the tortoise has moved further ahead.

Step 2: Setting up the Mathematical Model

Create a recursive sequence to describe the problem. If the tortoise moves for every unit distance Achilles covers, then we see that when Achilles runs the headstart of 100 yards, the tortoise is 10 yards ahead. When Achilles runs an additional 10 yards, the tortoise moves 1 more yard ahead, and this pattern continues.

Step 3: Modeling the Series

We identify this as an infinite geometric series where Achilles needs to cover an infinite sum of distances to reach the tortoise. The sequence is: 100, 10, 1, 0.1, 0.01, etc. Determine the total distance Achilles needs to run using the sum of an infinite geometric series formula.

Step 4: Solving the Geometric Series

The formula for the sum of an infinite geometric series is \( S = \frac{a}{1-r} \), where \( a \) is the first term (100 yards), and \( r \) is the common ratio (0.1 since each term is 10% of the previous one). Applying this formula gives: \( S = \frac{100}{1-0.1} = \frac{100}{0.9} = \frac{1000}{9} \approx 111.11 \) yards.

Step 5: Conclusion

Achilles will need to run approximately 111.11 yards to reach the point where the tortoise is at that moment. Therefore, Achilles will catch the tortoise after running this calculated distance.

Key Concepts

Infinite Geometric Series

In mathematics, an infinite geometric series is a series of numbers that continues indefinitely, where each term is a constant multiple of the previous term. This concept is at the heart of understanding Zeno's paradox, specifically in the scenario where Achilles tries to catch the tortoise. Here, each additional piece of distance Achilles runs is represented by a successive term in a geometric sequence. For Achilles chasing the tortoise:

• The first term, \(a\), is the initial 100 yards Achilles runs.
• Each subsequent distance is only 10% of the previous distance, giving us the common ratio \(r = 0.1\).

Using these values, we can calculate the total distance Achilles must run by taking the sum of this series, which is given by the formula: \( S = \frac{a}{1-r} \). Here, \(S\) is the total sum, while \(a\) and \(r\) are known values. In our case, it sums to approximately 111.11 yards. This series helps illustrate how, despite the seemingly infinite process, the sum converges to a specific number. This is why Achilles will indeed catch the tortoise, providing a resolution to the paradox.

Mathematical Modeling

Mathematical modeling allows us to translate real-world scenarios into mathematical expressions. In Zeno's paradox, we apply mathematical modeling to understand how Achilles can catch up to the tortoise. By considering the conditions of the race:

• Achilles runs ten times faster than the tortoise.
• The tortoise has a 100-yard head start.

We model this problem through a recursive sequence and identify it as a geometric series. By doing so, we create a clear mathematical pathway to solving the paradox. The model captures both the speed relation and the head start, turning them into a structured sequence of distances Achilles covers and where the tortoise is at each step. This mathematical approach demonstrates how seemingly impossible tasks can be broken down and solved with careful analysis and logical structuring.

Recursive Sequences

Recursive sequences are patterns where each term is derived from the previous term using a specific rule. In Zeno's paradox, we can describe the motion of Achilles and the tortoise using such sequences. Here, each time Achilles reaches the tortoise's previous position, the tortoise has moved forward a smaller segment, forming a recursive pattern:

• 100 yards (initial difference).
• 10 yards (next step).
• 1 yard, followed by 0.1 yards, and so forth.

This sequence continues indefinitely but generates diminishing returns, making it easier to calculate using the properties of recursive sequences. Though the intervals seem infinite, the sum of these distances converges thanks to the geometric nature of the sequence. Understanding recursive sequences helps us appreciate why the problem concludes that Achilles will catch the tortoise, even though the sequence appears endless.

Achilles and the Tortoise

"Achilles and the Tortoise" is a version of Zeno's paradox which illustrates how infinity can be found in seemingly finite and concrete situations. The paradox presents Achilles, a swift warrior, unable to surpass a slow-moving tortoise. Although at first glance, it seems Achilles should easily overtake the tortoise, the paradox introduces complexities:

• Every time Achilles reaches the tortoise's last position, the tortoise moves slightly forward.
• This movement creates smaller and smaller distances Achilles must cover.

By creating an endless series of steps where each is infinitesimally small compared to the previous, the paradox engages with infinite geometric series and recursive sequences. However, mathematics helps unlock the resolution: by summing an infinite series, we find that Achilles does indeed surpass the tortoise after approximately 111.11 yards. This paradox teaches how infinite series converge and deepen our understanding of limits and motion, neatly wrapped into this classical philosophical dilemma.