Problem 73 - Precalculus Mathematics for Calculus

Question

A ball is dropped from a height of \(80 \mathrm{ft}\). 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

On the fifth bounce, the ball rebounds to 25.3125 ft. Formula: \( a_n = 80 \times \left( \frac{3}{4} \right)^{n-1} \).

Step 1: Understand the Problem

We need to calculate how high the ball rebounds on the fifth bounce, given that it rebounds three-fourths of the distance fallen. Additionally, we need to find a formula for the rebound height on the nth bounce.

Step 2: Determine Rebound Height Formula

The rebound height forms a geometric sequence where each bounce is three-fourths of the previous height. Let's denote the initial height as \( a_1 = 80 \) ft. The rebound height for each subsequent bounce is three-fourths of the previous rebound height. For the nth bounce, the rebound height \( a_n \) is given by the formula:\[ a_n = 80 \times \left( \frac{3}{4} \right)^{n-1} \]

Step 3: Calculate the Fifth Bounce

Using the formula derived in Step 2, calculate the height the ball rebounds to on the fifth bounce:\[ a_5 = 80 \times \left( \frac{3}{4} \right)^{5-1} \] \[ a_5 = 80 \times \left( \frac{3}{4} \right)^4 \] \[ a_5 = 80 \times \frac{81}{256} \] \[ a_5 = 25.3125 \]ftThus, the ball rebounds to a height of 25.3125 ft on the fifth bounce.

Key Concepts

Understanding Rebound Height

When a ball is dropped from a certain height, it falls to the ground and then bounces back up. The height it reaches after bouncing is called the rebound height. This height depends on several factors, such as the material of the ball and its elasticity. In our example, we started with a ball dropped from 80 feet. On the first bounce, the ball rebounds to three-fourths of the distance it initially fell. This means the rebound height is determined by multiplying the initial height by the rebound ratio, which is 0.75 in this exercise. Understanding rebound height is crucial because it allows us to predict how the ball behaves over several bounces. The more bounces we have, the pattern of this rebound height forms a geometric sequence, which simplifies our ability to make calculations with repetitive formulas.

Elasticity of Materials

The elasticity of a ball is an essential factor in determining how high it will bounce. This property refers to the tendency of a material to return to its original shape after being deformed, such as when a ball hits the ground. A ball with high elasticity will rebound higher, indicating that it can store and return more of the energy it gained from the fall. In the given problem, the ball's elasticity is defined by the fact that it rebounds up to three-fourths of its falling height. This fractional rebound height derives from the ball's material properties, which could be rubber or a similar elastic material. Since elasticity affects the sequence of bounces, knowing this value is critical for calculating each successive rebound height accurately. Elasticity not only affects sports applications but also engineering fields where energy absorption and return are important.

Developing a Mathematical Formula

In exercises like this one, developing a mathematical formula becomes necessary to solve problems efficiently. Once we understand how elasticity impacts rebound height, we can model this behavior into a mathematical equation. The ball's rebound heights follow a geometric sequence where with each bounce, the height is determined by multiplying the previous height by a fixed ratio, the elasticity ratio (in this case, 0.75). The formula to calculate the rebound height on the nth bounce can be given by:\[ a_n = 80 \times \left( \frac{3}{4} \right)^{n-1} \]Here, 80 is the initial drop height, and \( \left( \frac{3}{4} \right)\) is the elasticity ratio repeated n-1 times. Using this formula, anyone can easily find how high the ball rebounds at any specific bounce number, simplifying predictive calculations over repeated trials.