Chapter 9.8, Problem 4CYP

To determine

A graph to represent the height of the ball after each bounce

Answer to Problem 4CYP

The sequence is $16,11.2,7.84,5.488,\mathrm{...}$

The graph is below

  

Explanation of Solution

Given information:

A tennis ball is dropped from a height of 12 feet and each bounce back 80% the height of the previous bounce

The ball bounces back to a height of 80% of its previous height, then $r=0.8$

And $a_1=12$

Concept Used:

The equation for nth term is $a_n=a_1{\left(r\right)}^{n-1}$

The ratio is given by the formula

  $r=\frac{a_2}{a_1}=\frac{a_3}{a_2}$

Calculation:

In order to find the sequence of each bounce of ball is as shown below,

  $\begin{array}{c}{a}_n=a_1{\left(r\right)}^{n-1}\\ a_1=12\\ r=0.8\\ \text{equation for nth term is}\\ a_n=12{\left(0.8\right)}^{n-1}\\ a_2=1{\left(0.8\right)}^{2-1}\\ =12{\left(0.8\right)}^1\\ a_2=12\times 0.8\\ a_2=9.6\\ a_3=12{\left(0.8\right)}^{3-1}\\ =12{\left(0.8\right)}^2\\ =12\times 0.64\\ =7.68\\ a_4=12{\left(0.8\right)}^{4-1}\\ =12{\left(0.8\right)}^3\\ =12\times 0.512\\ =6.144\\ a_5=12{\left(0.8\right)}^{5-1}\\ =12{\left(0.8\right)}^4\\ =12\times 0.4096\\ =4.915\end{array}$

The sequence is $12,9.6,7.68,6.14,4.91,\mathrm{...}$

The graph is below