Chapter 12.3, Problem 88AYU

To determine

To find:

a. The height of the ball bounce up after it strikes the ground for the third time.

Answer to Problem 88AYU

a. The height of the ball bounce up after it strikes the ground for the third time $=19.2$ inches

Explanation of Solution

Given:

It is given that the ball is dropped from a height of 30 feet and on each successive strike the height of the bounce is $0.8$ times the previous bounce height.

The height of the first strike is 30 inches.

The height of the second strike is $30\left(0.8\right)$ inches.

The height of the third strike is $30{\left(0.8\right)}^2$ inches.

The height of ball strikes sequence will be $30, 30\left(0.8\right), 3{\left(0.8\right)}^2, 30{\left(0.8\right)}^3,\cdots$

Therefore, the height of the ball bounces follows the geometric sequence.

a. The height of the ball bounce up after it strikes the ground for the third time.

Let’s find the $a_3=\left(30\right){\left(0.8\right)}^2=19.2$ inches.

To determine

To find:

b. The height of the bounce after it strikes the ground for $n$th time.

Answer to Problem 88AYU

b. The height of the bounce after it strikes the ground for nth time $a_n=\left(30\right){\left(0.8\right)}^{n-1}$.

Explanation of Solution

Given:

It is given that the ball is dropped from a height of 30 feet and on each successive strike the height of the bounce is $0.8$ times the previous bounce height.

The height of the first strike is 30 inches.

The height of the second strike is $30\left(0.8\right)$ inches.

The height of the third strike is $30{\left(0.8\right)}^2$ inches.

The height of ball strikes sequence will be $30, 30\left(0.8\right), 3{\left(0.8\right)}^2, 30{\left(0.8\right)}^3,\cdots$

Therefore, the height of the ball bounces follows the geometric sequence.

b. The height of the bounce after it strikes the ground for $n$th time.

As the sequence is geometric, the $n$th term must be ,

$a_n=a_1.r^{n-1}=\left(30\right){\left(0.8\right)}^{n-1}$

To determine

To find:

c. The number of times it has to strike the ground before its bounce is less than 6 inches.

Answer to Problem 88AYU

c. At the 9th strike, the height of the bounce is less than 6 inches.

Explanation of Solution

Given:

It is given that the ball is dropped from a height of 30 feet and on each successive strike the height of the bounce is $0.8$ times the previous bounce height.

The height of the first strike is 30 inches.

The height of the second strike is $30\left(0.8\right)$ inches.

The height of the third strike is $30{\left(0.8\right)}^2$ inches.

The height of ball strikes sequence will be $30, 30\left(0.8\right), 3{\left(0.8\right)}^2, 30{\left(0.8\right)}^3,\cdots$

Therefore, the height of the ball bounces follows the geometric sequence.

c. The number of times it has to strike the ground before its bounce is less than 6 inches.

The height of the bounce in the $n$th strike is $\left(30\right){\left(0.8\right)}^{n-1}$. For this to be exactly 6 inches requires that $a_n=1$.

$\left(30\right){\left(0.8\right)}^{n-1}=6$

Divide both sides by 30.

${\left(0.8\right)}^{n-1}= \frac{1}{5}$

Taking logarithm on both sides, $n-1={\log }_{0.8}\frac{1}{5}$.

$n=1+\frac{\log \frac{1}{5}}{\log 0.8}$

$n=1+\frac{-0.699}{-0.097}$

$n=1+7.21$

$n=8.21$

$n\approx 9$

At the 9th strike, the height of the bounce is less than 6 inches.

To determine

To find:

d. The total vertical distance travelled the ball before it stops.

Answer to Problem 88AYU

d. The total vertical distance travelled the ball before it stops $=\text{ 150}$ inches.

Explanation of Solution

Given:

It is given that the ball is dropped from a height of 30 feet and on each successive strike the height of the bounce is $0.8$ times the previous bounce height.

The height of the first strike is 30 inches.

The height of the second strike is $30\left(0.8\right)$ inches.

The height of the third strike is $30{\left(0.8\right)}^2$ inches.

The height of ball strikes sequence will be $30, 30\left(0.8\right), 3{\left(0.8\right)}^2, 30{\left(0.8\right)}^3,\cdots$

Therefore, the height of the ball bounces follows the geometric sequence.

d. The total vertical distance travelled the ball before it stops.

Ball stops its strikes means it converges to a particular point.

Sum of the infinite geometric series formula has to be used to find the total height travelled by the ball.

Convergence of an infinite geometric series theorem states that ‘If $\left|r\right|<1$ converges. Its sum is ${\displaystyle \sum }_{k=1}^{\infty }{a}_1{r}^{k-1}= \frac{a_1}{1-r}$’.

Required total height $=\frac{30}{1-0.8}=\frac{30}{0.2}=150$ inches.