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# Geometric Sequences and Exponential Functions

## Identifying and using common ratios

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Practice Geometric Sequences and Exponential Functions

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Bouncing Ball
Teacher Contributed

## Real World Applications – Algebra I

### Topic

Bounce that ball!

### Student Exploration

In the concept for Geometric Sequences and Exponential Functions, you learned about representing a bouncing ball as a sequence and had to calculate the how high the ball bounces after a static number of bounces, or vice versa. In all of those examples, you were given the rebound ratio. What if you didn’t know what the rebound ratio was, but had an actual ball?

Let’s try to see if we can figure out what the rebound ratio is, given just a ball and a measuring device. The bigger the measuring device, the better! (You might want to find a yardstick, or a meter stick. A measuring tape would be the best!)

Our objective here is to find the rebound ratio, and then use that to find out how many bounces it would take for the ball to start look like it’s rolling on the ground. We can then actually try it to find out if what we calculated was true.

In my trial, I have a tennis ball, and I dropped it from a height of 200 cm. When measuring its height after the first bounce, I measured the height at 111 cm. Just using these two measurements, I can find the rebound ratio.

Using the geometric sequence formula, we have 111=200(r)(21)\begin{align*}111 = 200(r)^{(2-1)}\end{align*}. 200 represents the initial height, and (2, 111) represents the second height after the first bounce at 111 cm.

1111110.555=200(r)1=200×r=r\begin{align*}111 &= 200(r)^1\\ 111 &= 200\times r\\ 0.555 &= r\end{align*}

We now know that this ball has a 55.5% rebound ratio. We can fill in a table or calculate the height of the ball after each further bounce.

After calculating all of the values, I have:

1 200
2 111
3 61.605
4 34.190775
5 18.97588013
6 10.53161347
7 5.845045476
8 3.244000239
9 1.800420133
10 0.999233174
11 0.554574411
12 0.307788798
13 0.170822783

From this table, I can see that after the 11th\begin{align*}11^{th}\end{align*} bounce, the height of the ball should be less than half a centimeter. That’s pretty small! I’d say that the ball would start to look like it’s rolling by the 11th\begin{align*}11^{th}\end{align*} bounce.

### Extension Investigation

Try getting a ball and a measuring tool and find the rebound ratio. After finding the rebound ratio, calculate the number of bounces that it would take for it to look like the ball would start rolling. Then actually do it with a ball! Would you get the same answers? Why or why not? Do you think other factors could change your results? (Maybe things like the temperature of the room, the slant of the surface, or the materials that the ball’s composed of.)

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