### Student Page 3.2.2 Packaging Stacked Cups

Using a stack of at least 15 same-size cups, gather the data requested in Student Page Exercise 3.2.2.1. Share the data with the class. Find the average data for the groups that measured the same types of cups. Use the average data to complete Student Page Exercise 3.2.2.2–3.2.2.7.

Consider the second size of cups. Answer Student Page Exercise 3.2.2.8 for these cups in comparison with the smaller cups.

What other questions can you ask and answer?

A company manufactures disposable cups and rectangular cardboard cartons to package them. The cups come in several sizes. You job is to develop a mathematical model that will help determine the relationship between the inside height, \(h\text{,}\) of the carton and the number of cups, \(c\text{,}\) it will hold if the cups are to be stacked in the carton.

#### 1.

You have been provided two different types of disposable cups. For the small cup, complete Table 3.2.2.1 showing the relationship between the number of cups, \(c\text{,}\) in a stack and the total height in centimeters, \(h\text{,}\) of the stack. Show accuracy to the nearest tenth of a centimeter.

Number of Cups, \(c\) |
Height of Stack, \(h\text{,}\) in cm |
---|---|

1 | |

2 | |

3 | |

4 | |

5 | |

6 | |

\(c\) |

#### 2.

Graph the data. Label the axes showing the variables and scales.

#### 3.

Find a formula that allows you to predict the height of the stack of cups based on the number of cups.

#### 4.

Predict the height of a stack of:

##### (a)

10 cups

##### (b)

26 cups

#### 5.

What physical features of a cup are relevant to how high the cups stack? In terms of the physical features of a cup, write a general rule for the height of a stack based on these physical features.

#### 6.

Someone said, “If you double the number of cups in a stack, the height of the stack doubles.” Is this thinking right or wrong? Why do you think so?

#### 7.

Using the graph in Student Page Exercise 3.2.2.2, how many cups will cartons of the following heights hold?

##### (a)

30 cm

##### (b)

50 cm

#### 8.

Use the large cups.

##### (a)

How would the graph be different from the graph of the small cups in terms of steepness? Explain.

##### (b)

Sketch an approximate graph for the large cups on the same set of axes as the graph in Student Page Exercise 3.2.2.2.