Chapter 12.3, Problem 86AYU

(a)

To determine

To find: The height of the ball bounce up to after its strike the ground for the 3^rd time.

Answer to Problem 86AYU

  $15.36\text{ feet}$

Explanation of Solution

Given: A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height.

  

Initial height of the ball = 30 feet.

After each bounces up loose one-fifth height of the previous one.

After 1 bounce height of ball $=0.8\times 30=24\text{ feet}$

After 2 bounce height of ball $=0.8\times 24=19.2\text{feet}$

After 3 bounce height of ball $=0.8\times 19.2=15.36\text{feet}$

(b)

To determine

To find: The height of the ball bounce up to after its strike the ground for the nth time.

Answer to Problem 86AYU

  $30{\left(0.8\right)}^n\text{feet}$

Explanation of Solution

Given: A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height.

  

Initial height of the ball, $a_0=30\text{ feet}$ .

After each bounces up loose one- fifth height of the previous one.

After 1 bounce height of ball $,a_1=0.8\times a_0=24\text{ feet}$

After 2 bounce height of ball $,a_2=0.8\times a_1={\left(0.8\right)}^2{\text{a}}_0\text{=19}\text{.2 feet}$

After n bounce height of ball $,a_n={\left(0.8\right)}^n\times a_0=30{\left(0.8\right)}^n\text{feet}$

(c)

To determine

To find: The number of bounce when height is less than 6 inches.

Answer to Problem 86AYU

  $7$

Explanation of Solution

Given: A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height.

  

Initial height of the ball, $a_0=30\text{ feet}$ .

After each bounces up loose one- fifth height of the previous one.

After n bounce height of ball $,a_n={\left(0.8\right)}^n\times a_0=30{\left(0.8\right)}^n\text{feet}$

So, $a_n<6$

  $\begin{array}{c}30{\left(0.8\right)}^n<6\\ {\left(0.8\right)}^n<0.2\\ n<\frac{\ln 0.2}{\ln 0.8}\\ n<7.2\end{array}$

Hence, 7 bounces.

(d)

To determine

To find: The total distance covered when stops bouncing.

Answer to Problem 86AYU

  $270\text{ feet}$

Explanation of Solution

Given: A ball is dropped from a height of 30 feet. Each time it strikes the ground, it bounces up to 0.8 of the previous height.

  

Initial height of the ball, $a_0=30\text{ feet}$ .

After each bounces up loose one- fifth height of the previous one.

After n bounce height of ball $,a_n={\left(0.8\right)}^n\times a_0=30{\left(0.8\right)}^n\text{feet}$

Total distance travelled by ball $,S_n=a_0+2a_1+2a_2+\mathrm{.............}\infty$

  $\begin{array}{l}{S}_n=30+2\left(0.8\times 30+{\left(0.8\right)}^2\times 30+\mathrm{.........}\right)\\ S_n=30+2\times 0.8\times \frac{30}{1-0.8}\\ S_n=270\text{ feet}\end{array}$

Hence, total distance travel by ball before its stops is $270\text{ feet}$ .