# Lesson 21

Predicting Populations

Let's use linear and exponential models to represent and understand population changes.

### Problem 1

The table shows the height of a ball after different numbers of bounces.

\(n\) | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

\(h\) | 83 | 61 | 46 | 35 | 26 |

- Can the height, \(h\), in centimeters, after \(n\) bounces be modeled accurately by a linear function? Explain your reasoning.
- Can the height, \(h\), after \(n\) bounces be modeled accurately by an exponential function? Explain your reasoning.
- Create a model for the height of the ball after \(n\) bounces and plot the predicted values with the data.
- Use your model to estimate the height the ball was dropped from.
- Use your model to estimate how many bounces it takes before the rebound height is less than 10 cm.

### Problem 2

Mai used a computer simulation to roll number cubes and count how many rolls it took before all of the cubes came up sixes. Here is a table showing her results.

number of cubes \(d\), |
1 | 2 | 3 | 4 |
---|---|---|---|---|

number of rolls \(r\), |
5 | 31 | 143 | 788 |

Would a linear or exponential function be appropriate for modeling the relationship between \(d\) and \(r\)? Explain how you know.

### Problem 3

A ramp is two meters long. Priya wants to investigate how the distance a basketball rolls is related to the location on the ramp where it is released.

Recommend a way Priya can gather data to help understand this relationship.

### Problem 4

Here are the graphs of three functions.

Which of these functions decays the most quickly? Which one decays the least quickly?

### Problem 5

The bungee jump in Rishikesh, India is 83 meters high. The jumper free falls for 5 seconds to about 30 meters above the river.

- Draw a graph of the bungee jump in Rishikesh.
- Identify and describe three pieces of important information you can learn from the graph of the bungee jump.