Chapter 7.1, Problem 33PPSE

Solution

To Write: The exponential function that models for the bouncing ball.

The model for the bouncing ball is $y=57.4{\left(0.61\right)}^x$ .

Given:

Ball drops from $=5ft$

First bounce height $=35$ inches

Second bounce height $=21.5$ inches

Explanation:

By using the heights of the first and second bounces, determine $r$ . According to the referee,

the first bounce measured $35$ inches, while the second bounce measured $21.5$ inches.

Substitute $y_1=35$ and $y_2=21.5$ in the formula for $r=\frac{y_2-y_1}{y_1}$ :

  $\begin{array}{c}r=\frac{21.5-35}{35}\\ =\frac{-13.5}{35}\\ \approx -0.39\end{array}$

Find the growth or decay factor $b$ :

Use the formula $b=1+r$ to determine $b$ .

Substitute $r=-0.39$ in $b=1+r$ :

  $\begin{array}{c}b=1+\left(-0.39\right)\\ =0.61\end{array}$

Apply the model $y=a{b}^x$ .

In the first bounce, tennis ball rebound was approximately $35$ inches, so the initial values are $x=1$ and $y=35$ .

Solve $a$ by using $x=1,b=0.61$ and $y=35$ in the equation $y=a{b}^x$ :

  $\begin{array}{c}35=a{\left(0.61\right)}^1\\ 35=a\left(0.61\right)\\ a=57.4\end{array}$

Now, substitute $a=57.4$ and $b=0.61$ in the equation $y=a{b}^x$ :

  $y=57.4{\left(0.61\right)}^x$

Thus, the model for the bouncing ball is $y=57.4{\left(0.61\right)}^x$ .