Section 3.2 Solving Linear Equations: Packaging Stacked Cups
Subsection 3.2.1 Overview
As you have experienced, there are many examples from your life that are reasonably modeled by linear functions. In this lesson, you will consider a packaging problem. As you solve the problem, think of other items of similar size that you stack and store.
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.
Homework 3.2.3 Homework
1.
Graph Packaging Stacked Cups average class data and your group data.
(a)
On Desmos, type the class data into a table, then type, \(y = mx + b\text{.}\) Choose ‘both’ to set up sliders for \(m\) and \(b\text{.}\) Then play with both \(m\) and \(b\) until you find a line that seems to best fit the data. Record the equation for the line.
(b)
Repeat Task 3.2.3.1.a for your group's data. Some equations exactly fit the data. If this is the case for your group, also find the equation algebraically.
(c)
Compare the equations you found for the average class data and for your group data. Which one do you think is more accurate? Why?
(d)
Save your work on this problem. It will be revisited in a later lesson.
2.
Data for Packaging Stacked Cups from a previous class is listed in Table 3.2.3.1. The table shows the relationship between the number of cups, \(c\text{,}\) in a stack and the total height in centimeters, \(h\text{,}\) of the stack with accuracy to the nearest tenth of a centimeter.
| Number of Cups, \(c\) |
Height of Stack, \(h\text{,}\) in cm by Group Number |
||||||
|---|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | 6 | Average | |
| 1 | 7 | 7 | 7.2 | 7 | 7.4 | 7 | |
| 2 | 7.5 | 7.5 | 7.6 | 7.5 | 7.9 | 7.5 | |
| 3 | 8 | 8 | 8.1 | 8 | 8.1 | 8 | |
| 4 | 8.5 | 8.5 | 8.6 | 8.5 | 8.6 | 8.5 | |
| 5 | 9 | 9 | 9 | 9 | 9.1 | 9.5 | |
| 6 | 9.5 | 9.5 | 9.5 | 9.5 | 9.7 | 9.5 | |
| \(c\) | |||||||
(a)
In the table above, which set of data is perfectly linear? How do you know?
(b)
Find an equation to fit the linear data you chose in Task 3.2.3.2.a (see Task 3.2.3.1.a). Explain both the slope and \(y\)-intercept in terms of stacked cups.
(c)
Use the equation you found to determine the internal height of a package that will contain 50 cups. Show and explain your work.
(d)
Use the equation you found to determine the number of stacked cups that will fit inside a package with internal height of 100 cm. Show and explain your work.
(e)
Find the average class data for the data Table 3.2.3.1. Fit an equation to the data (see Task 3.2.3.1.a).
(f)
Solve Task 3.2.3.2.c and Task 3.2.3.2.d using the equation in Task 3.2.3.2.e.
3.
You pay a $5 entrance fee and $3 per ride at a fair.
(a)
Find an equation that fits this situation. How do you know your equation is correct?
(b)
You want to go on 7 rides at the fair (see Task 3.2.3.1.a). How much money do you need to bring along?
(c)
You go to the fair with $15 in your pocket. How many rides can you ride?
(d)
Solve the equation in Task 3.2.3.3.a for the independent variable. What does this new equation mean?
4.
Solve each of the following equations for \(x\text{,}\) the independent variable.
(a)
\(y = 3x - 4\)
(b)
\(y = \frac{1}{2}x + 6\)
(c)
\(y = -4.2x + 5\)