Problem 82 - Algebra and Trigonometry

Question

Bungee Jumping A bungee jumper plummets from a high bridge to the river below and then bounces back over and over again. At time \(t\) seconds after her jump, her height \(H\) (in meters) above the river is given by \(H(t)=100+75 e^{-t / 20} \cos \left(\frac{\pi}{4} t\right) .\) Find her height at the times indicated in the table.

Step-by-Step Solution

Answer

Calculate \( H(t) \) using the function for each specified \( t \) in the table.

Step 1: Understand the Function

The height function is given by \( H(t) = 100 + 75 e^{-t / 20} \cos \left(\frac{\pi}{4} t\right) \). This function represents the height above the river in meters at any given time \( t \), where the initial height is 100 meters, and the oscillation term \( 75 e^{-t/20} \cos \left(\frac{\pi}{4}t\right) \) models the bouncing effect as it diminishes over time.

Step 2: Evaluate the Function at Given Times

We will calculate the height \( H(t) \) at specific times from the table. For each specified \( t \), substitute into the formula:- For \( t = t_1 \): Substitute \( t_1 \) into \( H(t) \).- For \( t = t_2 \): Substitute \( t_2 \) into \( H(t) \).- Continue this process for each time value in the table.

Step 3: Calculation for Example Time Value

Let's exemplify with one value, say \( t = 0 \):For \( t = 0 \), substitute into the height equation:\[ H(0) = 100 + 75 e^{-0 / 20} \cos \left(\frac{\pi}{4} \times 0\right) \]This simplifies to:\[ H(0) = 100 + 75 \times 1 \times 1 = 175 \,\text{meters} \]

Step 4: Interpret the Results

The calculated heights represent how the bungee jumper's height changes over time as a result of the damping oscillation modeled by the function. Substitute and calculate each required time value from the table to complete the findings.

Key Concepts

Damping Oscillation

In this context, damping oscillation refers to the way the jumper's bounce decreases over time due to the effects of both gravity and the resistance from the bungee cord. The oscillatory motion is created by the function within the height formula: - The term \(\cos \left(\frac{\pi}{4} t\right)\) represents a wave-like oscillation pattern. This is because the cosine function regularly fluctuates between -1 and 1, creating peaks and troughs. - The damping effect comes from \(75 e^{-t/20}\), which causes these oscillations to reduce in magnitude as time progresses.

• Initially, the oscillation is strong, with a high impact on the height of the jumper.
• As \(t\) increases, the factor \(e^{-t/20}\) becomes smaller, leading to smaller jumps.

This damping mechanism ensures that the bounces will gradually decrease, providing a realistic model of what happens during an actual bungee jump.

Height Function

The height function \(H(t) = 100 + 75 e^{-t / 20} \cos \left(\frac{\pi}{4} t\right)\) is a mathematical representation of the jumper's height above the river over time. Let's break down its components: - **Initial Height:** The constant term "100" in the equation signifies the highest point from which the jumper starts descending. - **Oscillation Component:** The rest of the equation \(75 e^{-t / 20} \cos \left(\frac{\pi}{4} t\right)\) models how the height changes in a back-and-forth motion during the jump. This formula captures both the periodic bouncing and the decreasing height of the jumper as the session continues. By substituting different time values into the function, you can determine the exact height at specific moments, providing insight into how the motion evolves. Regular substitution of time values into this function will clearly demonstrate how quickly or slowly the height adjustment occurs during the jump.

Exponential Decay

Exponential decay in this scenario represents how the amplitude of the bungee jumper's oscillations reduces over time. The term \(e^{-t/20}\) in the height function governs this decay. Here's how it works: - The exponential part \(e^{-t/20}\) is a decreasing function. As time \(t\) increases, the value of \(e^{-t/20}\) becomes smaller. - Initially, the effect on the oscillation is significant, but as time goes on, the influence diminishes quickly.

• At \(t=0\), the bungee jumper experiences the maximum height alteration. The decay starts strong.
• As time progresses, this decay means less height fluctuation on each bounce until it eventually stabilizes.

This characteristic of exponential decay mirrors real-world phenomena quite effectively, such as how energy dissipates over time, resulting in less pronounced oscillations.