Problem 89 - Precalculus: Mathematics for Calculus

Question

A ball is dropped from a height of 80 ft. The elasticity of this ball is such that it rebounds three-fourths of the distance it has fallen. How high does the ball rebound on the fifth bounce? Find a formula for how high the ball rebounds on the \(n\) th bounce.

Step-by-Step Solution

Answer

The ball rebounds to 18.984375 ft on the fifth bounce. The general formula for the \( n \)-th bounce is \( h_n = 80 \times \left(\frac{3}{4}\right)^n \).

Step 1: Initial Drop Height

The problem states that the ball is initially dropped from a height of 80 ft, so the initial height is 80 ft.

Step 2: Understanding the Rebound Mechanism

Each time the ball falls, it rebounds to three-fourths of the distance it has fallen. This means if the ball falls a certain height, the height of the next rebound is \( \frac{3}{4} \) times the previous fall height.

Step 3: Compute First Bounce Height

For the first bounce, the ball rebounds to \( \frac{3}{4} \times 80 \ = 60 \) ft.

Step 4: Compute Second Bounce Height

From the 60 ft height, the ball falls and rebounds \( \frac{3}{4} \times 60 \ = 45 \) ft.

Step 5: Compute Third Bounce Height

Now from the 45 ft height, the ball rebounds to \( \frac{3}{4} \times 45 \ = 33.75 \) ft.

Step 6: Compute Fourth Bounce Height

For the fourth bounce, the rebound height is \( \frac{3}{4} \times 33.75 \ = 25.3125 \) ft.

Step 7: Compute Fifth Bounce Height

For the fifth bounce, the rebound height is \( \frac{3}{4} \times 25.3125 \ = 18.984375 \) ft.

Step 8: Formula for nth Bounce Height

The rebound height formula for the \( n \)-th bounce can be expressed as: \[ h_n = 80 \times \left(\frac{3}{4}\right)^n \] where \( h_n \) is the height of the \( n \)-th rebound.

Key Concepts

Rebound Height Calculation

A rebound height calculation involves determining how high an object bounces back after being dropped from a certain height. In this particular scenario, a ball is dropped from an initial height of 80 feet. Each time it hits the ground, it reaches a maximum rebound height that is three-fourths of its previous falling height.
To calculate the rebound height for each bounce, you need to follow a sequential approach:

• Start with the initial drop height, which is given as 80 feet.
• For each subsequent bounce, multiply the maximum height of the previous bounce by the rebound fraction, which is three-fourths (or 0.75).

For example, after the first impact with the ground, the ball rebounds to 60 feet, which is calculated as \( \frac{3}{4} \times 80 \). This process is repeated for each bounce. The bounce heights decrease as the number of bounces increases, following the given elasticity condition of the ball.

Geometric Series Formula

The rebound height of the ball can be described using the geometric series formula. This formula applies to sequences where each term is found by multiplying the previous term by a constant ratio. In the context of our problem, this constant ratio is \( \frac{3}{4} \).
The general expression for the nth term in a geometric sequence is given by:\[ a_n = a_1 \times r^{n-1} \] where:

• \( a_1 \) is the first term (initial height), which is 80 feet in this case.
• \( r \) is the common ratio, equal to \( \frac{3}{4} \).
• \( n \) is the term number.

For the rebound heights of the ball, the formula becomes:\[ h_n = 80 \times \left(\frac{3}{4}\right)^n \] This formula allows calculation of the rebound height on any bounce. By substituting the desired bounce number for \( n \), students can solve for the precise rebound height.

Precalculus Problem Solving

Precalculus problem solving often involves working with sequences, series, and functions, just like in this exercise. Understanding how to derive and apply formulas is key.
To tackle such problems:

• Identify the initial conditions: Know what you start with, like the initial drop height of the ball.
• Recognize patterns or sequences: Here, the problem involves a geometric sequence due to the consistent rebound ratio.
• Apply mathematical formulas: Such as the geometric series formula to compute the nth rebound height.

These steps are crucial in preparatory mathematics courses like precalculus. They help lay the groundwork for exploring more complex mathematical concepts and real-world problem-solving. Understanding the step-by-step logical progression from initial conditions through sequence analysis to applying formulas enhances comprehension and retention of mathematical principles.