# Lesson 2

Patterns of Growth

Let’s compare different patterns of growth.

### Problem 1

A population of ants is 10,000 at the start of April. Since then, it triples each month.

- Complete the table.
- What do you notice about the population differences from month to month?
- If there are \(n\) ants one month, how many ants will there be a month later?

months since April | number of ants |
---|---|

0 | |

1 | |

2 | |

3 | |

4 |

### Problem 2

A swimming pool contains 500 gallons of water. A hose is turned on, and it fills the pool at a rate of 24 gallons per minute. Which expression represents the amount of water in the pool, in gallons, after 8 minutes?

\(500 \boldcdot 24 \boldcdot 8\)

\(500 + 24 + 8\)

\(500 + 24 \boldcdot 8\)

\(500 \boldcdot 24^8\)

### Problem 3

The population of a city is 100,000. It doubles each decade for 5 decades. Select **all** expressions that represent the population of the city after 5 decades.

32,000

320,000

\(100,\!000 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2 \boldcdot 2\)

\(100,\!000 \boldcdot 5^2\)

\(100,\!000 \boldcdot 2^5\)

### Problem 4

The table shows the height, in centimeters, of the water in a swimming pool at different times since the pool started to be filled.

- Does the height of the water increase by the same amount each minute? Explain how you know.
- Does the height of the water increase by the same factor each minute? Explain how you know.

minutes | height |
---|---|

0 | 150 |

1 | 150.5 |

2 | 151 |

3 | 151.5 |

### Problem 5

Bank account C starts with $10 and doubles each week. Bank account D starts with $1,000 and grows by $500 each week.

When will account C contain more money than account D? Explain your reasoning.

### Problem 6

Suppose \(C\) is a rule that takes time as the input and gives your class on Monday as the output. For example, \(C(\text{10:15}) = \text{Biology}\).

- Write three sample input-output pairs for \(C\).
- Does each input to \(C\) have exactly one output? Explain how you know.
- Explain why \(C\) is a function.

### Problem 7

The rule that defines function \(f\) is \(f(x) = x^2+1\). Complete the table. Then, sketch a graph of function \(f\).

\(x\) | \(f(x)\) |
---|---|

-4 | 17 |

-2 | |

0 | |

2 | |

4 |

### Problem 8

The scatter plot shows the rent prices for apartments in a large city over ten years.

- The best fit line is given by the equation \(y=134.02x+655.40\), where \(y\) represents the rent price in dollars, and \(x\) the time in years. Use it to estimate the rent price after 8 years. Show your reasoning.
- Use the best fit line to estimate the number of years it will take the rent price to equal $2,500. Show your reasoning.