Chapter 9.4, Problem 39E

Solution

To calculate: The total distance traveled by the ball until it stop bouncing if a rubber ball is dropped from height of 2m. The rubber ball could bounce to 90 percent of the height from which they were dropped.

The resultant total distance is 38m.

Given information:

A rubber ball is dropped from height of 2m. The rubber ball could bounce to 90 percent of the height from which they were dropped.

Formula used: The sum of a infinite geometric series is: $Sum=\frac{a}{1-r}$

Calculation:

A ball is dropped from a height of 2m.

Assume that the initial height is: x

Then the next bounce height is: $0.9x$

Now the distance traveled can be modeled by the series:

  $\begin{array}{c}L=x+2\left({\left(0.9\right)}^{1}x+2{\left(0.9\right)}^{2}x+\cdots \right)\\ =x\left[1+2\left({\left(0.9\right)}^1+2{\left(0.9\right)}^2+\cdots \right)\right]\end{array}$

Substitute the values in the sum formula $Sum=\frac{a}{1-r}$ ,

  $\begin{array}{c}L=x\left[1+2\left(\frac{0.9}{1-0.9}\right)\right]\\ =x\left[1+2\left(\frac{0.9}{0.1}\right)\right]\\ =x\left[1+2\left(9\right)\right]\\ =19x\end{array}$

The value of x is 2. Then

  $\begin{array}{c}19x=19\left(2\right)\\ =38\end{array}$

Therefore, the total distance traveled is 38m.