Problem 135 - Precalculus with Limits

Question

A bungee jumper is jumping off the New River Gorge Bridge in West Virginia, which has a height of \( 876 \) feet. The cord stretches \( 850 \) feet and the jumper rebounds \( 75\% \) of the distance fallen. (a) After jumping and rebounding \( 10 \) times, how far has the jumper traveled downward? How far has the jumper traveled upward? What is the total distance traveled downward and upward? (b) Approximate the total distance, both downward and upward, that the jumper travels before coming to rest.

Step-by-Step Solution

Answer

The total distance traveled downward in 10 jumps is \( 850 \times (1- (0.75^2)^{10}) / (1 - 0.75^2) \) feet, the total distance traveled upward in 10 jumps is \((0.75 \times 850) \times (1- (0.75^2)^{9}) / (1 - 0.75^2) \) feet, and the total distance traveled both ways together in 10 jumps is the sum of these two. The total distance traveled before coming to rest equals the sum of an infinite downward series plus the sum of an infinite upward series, which is \( 850 / (1 - 0.75^2) + 0.75*850 / (1 - 0.75^2) \) feet.

Step 1: Analyze the given information

From the problem, it is learned that the height of the bridge is \(876 \) feet, the jumper rebounds \( 75\% \) of the distance fallen which translates to \(0.75\) as fraction and for now, we're asked about the \(10\) times of jump.

Step 2: Total Distance traveled downward in 10 jumps

As the cord stretches only \(850 \) feet, the jumper in the first fall travels down \(850 \) feet. For the subsequent falls, he travels down only \(75\% \) of the distance he rebounded up in the previous jump. Hence, the total distance traveled downward in \(10 \) jumps makes a geometric series that starts from \(850\) feet, with a common ratio of \(0.75^2 \) (because each time the jumper falls again, he only travels \(75\% \) of the distance he rebounded up in the previous cycle.). Therefore, the total distance is \( 850 \times (1- (0.75^2)^{10}) / (1 - 0.75^2) \) feet.

Step 3: Total distance traveled upward in 10 jumps

For the first jump, the jumper didn't travel upward. From the second jump onwards, the jumper travels upward by \((0.75)\) of the distance he traveled downward in the previous jump. Hence, the total distance traveled upward also forms a geometric series but starts from \((0.75 \times 850)\) feet with a common ratio of \((0.75^2)\). Therefore, the total distance traveled upward in 10 jumps is \((0.75 \times 850) \times (1- (0.75^2)^{9}) / (1 - 0.75^2) \) feet.

Step 4: Total distance traveled both ways together in 10 jumps

The total distance traveled both ways is the sum of the total distance traveled downward and the total distance traveled upward. Hence we simply need to add the results obtained in step 2 and 3.

Step 5: Total distance traveled before coming to rest

The jumper's downward and upward movement will continue till infinity theoretically, but practically, the movement may cease when the distance in one cycle becomes too small. As the ratios of both the sequences are less than 1, the sum of infinite terms of the geometric series equals \( a / (1 - r) \), where a is the first term and r is the common ratio. Thus, the total distance traveled before coming to rest equals the sum of the downward series with infinite terms plus the sum of the upward series with infinite terms, which can be calculated using the formula mentioned above.

Key Concepts

Infinite Geometric Series

When we talk about an infinite geometric series, we're discussing a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called a common ratio. For instance, if we take a look at the bungee jumper example, each rebound is 75% of the previous height, creating an infinite sequence if the jumper could rebound indefinitely.

One interesting aspect of infinite geometric series is that they can sum to a finite number if the common ratio is between -1 and 1. This might seem counterintuitive since we're adding up an infinite number of terms! In the bungee jumper's case, the common ratio is 0.75 squared because the jumper falls and then rebounds to 75% of that fall, effectively squaring the ratio. This value is between 0 and 1, indicating that our series will indeed converge. Mathematically, the sum of an infinite geometric series is expressed as \( S = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) the common ratio.

Series Convergence

The concept of series convergence is fundamental in understanding why the infinite geometric series doesn't just grow without bounds. A series converges when the sum of its terms approaches a certain value as the number of terms goes to infinity. Conversely, if the sum keeps getting larger without approaching a specific limit, the series is said to diverge.

For geometric series, the determining factor for convergence is the common ratio (\( r \)). If the common ratio's absolute value is less than 1, the series converges to a finite value. In the case of the bungee jumper, the common ratio applied to the series is \( r = 0.75^2 = 0.5625 \), which is indeed less than 1. Thus, we can say with certainty that the total distance traveled by the jumper converges to a finite number when an infinite number of jumps are considered.

Geometric Progression

A geometric progression is a sequence in which each term after the first is found by multiplying the previous term by a constant called the 'common ratio'. This progression can be visualized in our problem where each rebound is a fraction of the previous distance. The first term in the sequence is the first downward distance of 850 feet, and the common ratio is the square of 75% or 0.75.

The formula for finding the \(n\)th term of a geometric progression is \( t_n = a \times r^{(n-1)} \). When \(n\) becomes very large, the terms become very small if \(0 < r < 1\), which is why in the bungee jumper's scenario, the distances get smaller and smaller with each subsequent jump and eventually diminish, assuming an infinite number of jumps were possible.

Sum of Geometric Series

To understand the sum of a geometric series, it helps to have a formula which allows you to quickly find the sum of a finite or an infinite number of terms. For finite series, the formula is \( S_n = \frac{a(1-r^n)}{1-r} \) where \(n\) is the number of terms. For an infinite series, the formula simplifies to \( S = \frac{a}{1-r} \) as long as \| r \| < 1.

In the example of the bungee jumper, to find the total distance traveled after 10 jumps, we would use the first formula, plugging in the first downward distance as \(a\), and \(0.75^2\) as \(r\). For an infinite number of jumps, to get the total distance traveled before coming to rest, we would use the second formula. The beauty of this formula for infinite series is that it encapsulates the behavior of the entire sequence with just two parameters, \(a\) and \(r\), making complex problems like predicting the total distance a bungee jumper will travel quite manageable.