Examine Exponential Decay in Context

5.11.4 Examine Exponential Decay in Context

5.11.4 • Examine Exponential Decay in Context

Activity

The table shows some heights of a ball after a certain number of bounces.

Bounce Number	Height in Centimeters
0
1
2	73.5
3	51.5
4	36

1. Is this ball more or less bouncy than the tennis ball in activity 5.11.2, which has the equation $h = 150 \cdot (0.53)^{n}$ ? Explain or show your reasoning.

Compare your answer:

This ball is more bouncy. Each bounce of this ball reaches a height that is about 0.7 times that of the previous bounce $(51.5 \div 73.5 \approx 0.7)$ . The tennis ball in the earlier task loses about half of its height with each bounce.

2. From what height was the ball dropped? Explain or show your reasoning. Remember your units!

Compare your answer:

150 centimeters; $73.5 \div 0.7 = 105$ and $105 \div 0.7 = 150$ .

3. Write an equation that represents the bounce height of the ball, $h$ , in centimeters after $n$ bounces. Remember that an exponent may be entered using the "^" symbol.

Compare your answer:

$h = 150 \cdot (0.7)^{n}$

4. Which graph would more appropriately represent the equation for $h$ : Graph A or Graph B? Be prepared to show your reasoning.

A line graph shows height in centimeters decreasing rapidly with each bounce number, illustrating a curve that starts high and drops quickly, then levels out near zero. There are no plotted points.

A scatter plot shows height in centimeters decreasing with each bounce number, illustrating a rapid drop at first, then gradually leveling off as bounce number increases. Plots are available for each number of bounces.

Compare your answer:

Graph B is more appropriate. Only whole-number $n$ values make sense in this context. We cannot have bounce number 3.4, for example.

5. Will the $n^{th}$ bounce of this ball be lower than the $n^{th}$ bounce of the tennis ball with equation $h = 150 \cdot (0.53)^{n}$ ? Be prepared to show your reasoning.

Compare your answer:

No, the height of this ball after any given bounce will not be lower than that of the tennis ball because they were dropped from the same height and this ball has a greater rebound factor, so it will always bounce up to a higher point than the tennis ball.

6. The table shows an exponential decay that starts at 640. What exponential decay problem is it modeling? Use ^ to show your exponent.

$x$	0	1	2	3
$y$	640	160	40	10

Discrete x and corresponding y values, demonstrating an exponential decay relationship.

Compare your answer:

The pattern for the decay is starting at 640 and then decreasing by $\frac{3}{4}$ with only $\frac{1}{4}$ remaining. The equation is $y = 640 \cdot (1 - \frac{3}{4})^{x}$ or $y = 640 \cdot (\frac{1}{4})^{x}$ .

Self Check

The population of deer in a forest is divided in half every decade. Which of the following graphs could represent this situation?

Additional Resources

Writing Exponential Decay Functions

Example 1

Data has been entered into a table that shows exponential decay. How would you write the equation that represents this model?

$x$	0	1	2	3
$y$	200	100	50	25

Corresponding x and y values demonstrating an exponential decay relationship.

The formula for exponential decay is $y = a \cdot (1 - b)^{x}$ where a is the starting value and $b$ is the amount that value decreases each unit of time $x$ .

The pattern for the decay is starting at 200 and then decreasing by 12 with only 12 remaining. The equation is $y = 200 \cdot (1 - \frac{1}{2})^{x}$ or $y = 200 \cdot (\frac{1}{2})^{x}$ .

Example 2

A car sells for $4,000 and loses $\frac{1}{8}$ of its value each year. Write a function that gives the car’s value, $V (t)$ , $t$ years after it is sold.

Since the car loses $\frac{1}{8}$ of its value each year, it keeps $\frac{7}{8}$ of its value from year to year, so the growth factor is $\frac{7}{8}$ .

$V (t) = 4000 \cdot (\frac{7}{8})^{t}$

Try it

Writing Exponential Decay Functions

1. What exponential decay equation is modeled by the table of data given?

$x$	0	1	2	3
$y$	10,000	7,500	5,625	4,218.75

Table showing corresponding x and y values, illustrating an exponential decay relationship.

Compare your answer:

$y = 10,000 (1 - \frac{1}{4})$ or $y = 10,000 (\frac{3}{4})^{x}$ . The tennis ball in the earlier task loses about half of its height with each bounce.

A computer is purchased for $1,000 and loses $\frac{1}{5}$ of its value per year.

2. Write a function for the computer’s value, $V (t)$ , $t$ years after it is purchased.

Compare your answer:

Here is how to write the exponential function:

The starting amount is $1,000. The computer loses $\frac{1}{5}$ of its value each year, so it maintains $\frac{4}{5}$ of its value. The growth factor is $\frac{4}{5}$ .

The function is then $V (t) = 1000 \cdot (\frac{4}{5})^{t}$ .