Question

a-35. look back at the data given in problem a-18 that describes the rebound ratio for an official tennis ball. suppose you drop such a tennis ball from an initial height of 10 feet.
a. how high would it rebound after the first bounce?
b. how high would it rebound after the second bounce?
c. how high would it rebound after the fifth bounce?

Explanation:

Step1: Assume the rebound - ratio

Let's assume the rebound - ratio of an official tennis ball is $r$. Usually, the rebound - ratio of a tennis ball is about $\frac{2}{3}$. The height $h_n$ after $n$ bounces of an object dropped from an initial height $h_0$ is given by the formula $h_n=h_0\times r^n$. Here, $h_0 = 10$ feet.

Step2: Calculate the height after the first bounce

For $n = 1$, using the formula $h_n=h_0\times r^n$, we substitute $h_0 = 10$ and $r=\frac{2}{3}$.
$h_1=10\times\frac{2}{3}=\frac{20}{3}\approx6.67$ feet.

Step3: Calculate the height after the second bounce

For $n = 2$, we substitute $h_0 = 10$ and $r=\frac{2}{3}$ into the formula $h_n=h_0\times r^n$.
$h_2=10\times(\frac{2}{3})^2=10\times\frac{4}{9}=\frac{40}{9}\approx4.44$ feet.

Step4: Calculate the height after the fifth bounce

For $n = 5$, we substitute $h_0 = 10$ and $r=\frac{2}{3}$ into the formula $h_n=h_0\times r^n$.
$h_5=10\times(\frac{2}{3})^5=10\times\frac{32}{243}=\frac{320}{243}\approx1.32$ feet.

Answer:

a. $\frac{20}{3}\approx6.67$ feet
b. $\frac{40}{9}\approx4.44$ feet
c. $\frac{320}{243}\approx1.32$ feet