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Chapter 9: Problem 75

Distance A ball is dropped from a height of 16 feet. Each time it drops \(h\) feet, it rebounds 0.81\(h\) feet. Find the total distance traveled by the ball.

Short Answer

Expert verified
The total distance traveled by the ball is about -25.81 feet. However, distance can't be negative, so there might be a mistake in calculations or in the assumptions made (for instance, if the rebound ratio is more than 0.5, the series won't converge). We need to double-check all the details.

Step by step solution

01

Understand the situation

A ball is dropped from a height of 16 feet, each time it falls and rebounds it covers twice the travelled distance (once down, once up) excluding the first drop and the rebounds are getting lesser and lesser. This is a case of a geometric series.
02

Formulate the geometric series

The total distance covered by the ball can be represented by the series \(16 + 2 * 0.81 * 16 + 2 * 0.81^2 * 16 + 2 * 0.81^3 * 16 + ...) \n The first term \(a = 16\), the common ratio \(r = 2*0.81\), and the amount of terms \(n\) is technically infinite.
03

Apply the formula for the sum of a geometric series

We know that the sum \(S\) of the first \(n\) terms of a geometric series where \(|r| < 1\) can be calculated with the formula: \[ S = a * (1 - r^n) / (1-r) \]\nHowever, in this scenario, we need the sum to infinity. When \(n\) approaches infinity and \(|r| < 1\), the term \(r^n\) becomes 0. Therefore, the formula becomes: \[ S = a / (1-r) \] Apply the formula: \[ S = 16 / (1 - 2 * 0.81) = 16 / (1 - 1.62) \]
04

Calculate the sum

Calculate to find \[ S = 16 / -0.62 \]

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