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Exponential Growth and Decay Lab – AAT

Worth a Quiz grade

Names: ___________________________

____________________________

Materials Bag Number: _______

Save this document as: for the rest of the labs.

Grading Rubric –

**Activity: **

Practice Problems

Frogs in A Pond

Eagle Wildlife Refuge

Tennis Ball Rebound

**Points Possible **

You will use this document

10

**Points Earned **

25

25

25

Activity Turned in on Time:

Materials Returned in bag

**Total: **

10

5

**100 **

Exponential Growth and Decay

**Definition of the Exponential Growth/Decay **

𝑦 = 𝑎𝑏 𝑥

, where:

*y* is the population at any time

*a* is the starting population

*b* is a positive constant other than one

*x* is any number greater than or equal to zero. (time)

Exponential Growth when *b __greater than one_______ *

Exponential Decay when *b ___less than one__________*

Example:

A herd of buffalo began with a population of 100,000. The average birth rate has been 25% per year, but the death rate has been 48% per year. a) What is the difference between the birth rate and the death rate?

23% b) Is the population increasing or decreasing?

Decreasing, because the death rate is higher than birth rate. c) Write an exponential equation that models this situation, letting *y* is the population at any time and *x* is the time in years.

Y=100,000(.77)^x d) Assuming that these rates stay the same and there are no other factors, how long will it take the population to fall below 5,000? Round your answer to the nearest year.

5,000=100,000(.77)^x

0.05=(.77)^x

Ln(0.05)=ln(0.77)^x

Ln(.05)=xln(.77)

X=ln(0.05)/ln(.77) = 11.46 years

The population fell below 5,000 in the 12 th

year.

For all of these experiments, you will be using the GRAPH program on your laptop to enter data. You will type up the results of your experiments in this Word document, then drop it in the class Google Doc folder at the end of class.

1. Start the GRAPH program.

2. Function, Insert Point series

3. Enter the following data:

5

6

7

8

9

10

11

0

1

2

3

4

12

13

14

15

16

17

5

5

4

4

7

6

6

24

20

12

10

9

4

3

2

2

2

1

4. Hit OK.

5. Zoom, Fit to get a better graph of the data.

Your graph should look like this:

6. Function, Insert Trendline, Exponential, OK will get the exponential function that fits this curve. Your graph should now look like this:

7. Double-click on the the label of the x and y axis. button to change

In this experiment you will be using Skittles to model population growth. DO NOT EAT THE

SKITTLES UNTIL YOU HAVE FINISHED ALL THE INVESTIGATIONS!

Materials: a paper plate, Skittles, and a cup.

1. Put two Skittles in the cup and record this in the table below under 0. The

Skittles represent two frogs happily living in the local pond.

2. Shake the cup and pour the candy onto the paper plate. For every one with the s showing, add in another Skittle and record the new population (the number of Skittles on your plate.)

3. Continue this process and fill in the table.

4. Enter the data into the GRAPH program and make a scatterplot.

5. Calculate the exponential trendline for your data. Copy and paste the graph and your trendline equation here. Be sure to label your graph.

6. Using complete sentences, answer the following: a) How well you feel that the function fits your data? b) What is the rate of growth with this model? c) Using your equation predict what the population would be after 10 months. d) Using your equation, how long would it take for the population to reach 150 frogs?

Month: 0 1 2 3 4 5

# Frogs:

In this experiment you will be using Skittles to model population decay.

DO NOT EAT THE SKITTLES UNTIL YOU HAVE FINISHED ALL THE

INVESTIGATIONS!

Materials: a paper plate that has a shaded region and 25 Skittles in a cup.

1. The 25 Skittles represent the starting population of bald eagles in a wildlife refuge. Record this on the table under 0.

2. Shake the skittles onto the plate. Any Skittles that land in the shaded region must be removed from the plate as they have “died”. Count the number of Skittles that are left and record the number in the table below.

3. Continue this at least 5 times, or until the population is zero. If the population is zero, you cannot enter this value into the GRAPH program. It will not perform the exponential trendline if zero is a y-value.

4. Enter the data into the GRAPH program and make a scatterplot.

5. Calculate the exponential trendline for your data. Copy and paste the graph and your trendline equation here. Be sure to label your graph.

6. Using complete sentences, answer the following: a) How well you feel that the function fits your data? b) What is the rate of decay with this model? c) Using the equation, what would you predict the population to be after 7 years?

Year: 0 1 2 3 4 5

# Eagles:

In this experiment you will drop a tennis ball from a certain height and measure each rebound height.

Materials: meter stick or tape measure, tennis ball, chair or platform.

1. One person stands on a chair and holds the tennis ball out at a comfortable height. The tennis ball must be dropped from the same height every time.

2. Measure the height in cm. Record your results in the table below.

3. Drop the ball. One group member should catch the ball at its highest point after one bounce and hold it still. Another member measures the height. Record the data in the table.

4. Drop the ball from the same height and let it bounce twice before you catch and measure.

Repeat for 3, 4 ,5 bounces.

5. Enter the data into the GRAPH program and make a scatterplot.

6. Calculate the exponential trendline for your data. Copy and paste the graph and your trendline equation here. Be sure to label your graph.

7. Using complete sentences, answer the following: a) How well you feel that the function fits your data? b) What is the % decrease in bounce height? c) What would you predict the height would be after 7 bounces?

Number of bounces

Height, cm

0 1 2 3 4 5