Chapter 7.5, Problem 1MRPS

Solution

i.

To state: An infinite series to model the total distance traveled by the ball, excluding the distance traveled before the first bounce if a ball is dropped from a height of 12 feet. Each time the ball hits the ground, it bounces to 70% of its previous height.

The resultant infinite series is $d=12\left(0.7\right)+12{\left(0.7\right)}^2+\cdots$ .

Given information:

A ball is dropped from a height of 12 feet. Each time the ball hits the ground, it bounces to 70% of its previous height.

Explanation:

A ball is dropped from a height of 12 feet. It means the first term is $a_1=12$ .

Each time the ball hits the ground, it bounces to 70% of its previous height. Then the common ratio will become $r=0.7$ .

The nth term of geometric series is given by: $a_n=a_1{r}^{n-1}$

The second term is:

  $\begin{array}{c}{a}_2=a_1{r}^{2-1}\\ =12\left(0.7\right)\end{array}$

The third term is:

  $\begin{array}{c}{a}_3=12{\left(0.7\right)}^{3-1}\\ =12{\left(0.7\right)}^2\end{array}$

The distance will be equal to the sum of all the terms:

  $\begin{array}{l}d=a_1+a_2+a_3+\cdots \\ d=12+12\left(0.7\right)+12{\left(0.7\right)}^2+\cdots \end{array}$

The distance traveled before the first bounce must be excluded. Then distance traveled will become:

  $d=12\left(0.7\right)+12{\left(0.7\right)}^2+\cdots$

Therefore, the distance traveled by the ball can be represented in a infinite series $d=12\left(0.7\right)+12{\left(0.7\right)}^2+\cdots$ .

ii.

To calculate: The total distance traveled by the ball including the distance traveled before the first bounce.

The resultant distance is 40 feet.

Given information:

The series from part (a) is $d=12+12\left(0.7\right)+12{\left(0.7\right)}^2+\cdots$ .

Formula used: The sum formula of an infinite geometric series with first term $a_1=12$ and common ratio $r=0.7$ is given by $S=\frac{a_1}{1-r}$ . Where $\left|r\right|<1$ .

Calculation:

From part (a) the total distance is: $d=12+12\left(0.7\right)+12{\left(0.7\right)}^2+\cdots$

Use the sum formula: $S=\frac{a_1}{1-r}$

  $\begin{array}{c}S=\frac{12}{1-0.7}\\ =\frac{12}{0.3}\\ =40\end{array}$

Therefore, the sum of the series is 40 feet. It means the total distance traveled by the ball is 40 feet.