05.05 Modeling Exponential Behavior

Due No due date
Points 10
Questions 10
Time Limit None
Allowed Attempts Unlimited

Instructions

unit 5

Modeling Exponential Behavior

Goal

You will identify appropriate models for situations when given data and use exponential functions to model situations that involve exponential growth or decay. Then, apply that knowledge by answering questions.

Estimated completion time: 40 minutes

Watch

Sometimes data suggest an exponential relationship. For example, this table shows the bounce heights of a certain ball. We can see that the height decreases with each bounce.

To find out what fraction of the height remains after each bounce, we can divide two consecutive values: $LaTeX: \frac{61}{95}$ is about 0.642, $LaTeX: \frac{39}{61}$ is about 0.639, and $LaTeX: \frac{26}{39}$ is about 0.667.

All of these quotients are close to $LaTeX: \frac{2}{3}$ . This suggests that there is an exponential relationship between the number of bounces and the height of the bounce, and that the height is decreasing with a factor of about $LaTeX: \frac{2}{3}$ for each successive bounce.

Height of Ball bouncing
Bounce number	Bounce height in centimeters
1	95
2	61
3	39
4	26

The height h of the ball, in cm, after n bounces can be modeled by the equation:

$LaTeX: h=142\cdot\left(\frac{2}{3}\right)^n$

Here is a graph of the equation.

Graph with y-axis = height in centimeters and x-axis = number of bounces. 4 dots with no labels gradually decreasing.

This graph cannot be easily described. If you need an explanation of this image, please ask your teacher for help.

This graph shows both the points from the data and the points generated by the equation, which can give us new insights. For example, the height from which the ball was dropped is not given but can be determined. If $LaTeX: \frac{2}{3}$ of the initial height is about 95 centimeters, then that initial height is about 142.5 centimeters, because $LaTeX: 95\div\frac{2}{3}=142.5$ . For a second example, we can see that it will take 7 bounces before the rebound height is less than 10 centimeters.

Rate of change infographic and graph

Watch the following video about modeling exponential behavior.

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