Chapter 12, Problem 36RE

(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 36RE

  $8.44\text{ feet}$

Explanation of Solution

Given: A ball is dropped from a height of 20 feet. Each time it strikes the ground, it bounces up to three-quarters of the previous height.

Initial height of the ball = 20 feet.

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

After 1 bounce height of ball $=\frac{3}{4}\times 20=15\text{ feet}$

After 2 bounce height of ball $=\frac{3}{4}\times 15=11.25\text{ feet}$

After 3 bounce height of ball $=\frac{3}{4}\times 11.25=8.44\text{feet}$

(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 36RE

  $20{\left(0.75\right)}^n\text{feet}$

Explanation of Solution

Given: A ball is dropped from a height of 20 feet. Each time it strikes the ground, it bounces up to three-quarters of the previous height.

Initial height of the ball, $a_0=20\text{ feet}$ .

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

After 1 bounce height of ball $,a_1=\frac{3}{4}\times a_0=15\text{ feet}$

After 2 bounce height of ball $,a_2=\frac{3}{4}\times a_1={\left(\frac{3}{4}\right)}^2{\text{a}}_0\text{=11}\text{.25 feet}$

After n bounce height of ball $,a_n={\left(\frac{3}{4}\right)}^n\times a_0=20{\left(0.75\right)}^n\text{feet}$

(c)

To determine

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

Answer to Problem 36RE

  $4$

Explanation of Solution

Given: A ball is dropped from a height of 20 feet. Each time it strikes the ground, it bounces up to three-quarters of the previous height.

Initial height of the ball, $a_0=20\text{ feet}$ .

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

After n bounce height of ball $,a_n={\left(\frac{3}{4}\right)}^n\times a_0=20{\left(0.75\right)}^n\text{feet}$

So, $a_n<6$

  $\begin{array}{c}20{\left(0.75\right)}^n<6\\ {\left(0.75\right)}^n<0.3\\ n>\frac{\ln 0.3}{\ln 0.75}\\ n>4.18\end{array}$

Hence, 4 bounces.

(d)

To determine

To find: The total distance covered when stops bouncing.

Answer to Problem 36RE

  $140\text{ feet}$

Explanation of Solution

Given: A ball is dropped from a height of 20 feet. Each time it strikes the ground, it bounces up to three-quarters of the previous height.

Initial height of the ball, $a_0=20\text{ feet}$ .

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

After n bounce height of ball $,a_n={\left(\frac{3}{4}\right)}^n\times a_0=20{\left(0.75\right)}^n\text{feet}$

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

  $\begin{array}{l}{S}_n=20+2\left(\frac{3}{4}\times 20+{\left(\frac{3}{4}\right)}^2\times 20+\mathrm{.........}\right)\\ S_n=20+2\times \frac{3}{4}\times \frac{20}{1-\frac{3}{4}}\\ S_n=140\text{ feet}\end{array}$

Hence, total distance travel by ball before its stops is $140\text{ feet}$ .