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Let's break down the problem step by step:

### Data Table:
The table provides the number of cups and the corresponding height of the stack in centimeters.

| Number of Cups (c) | Height of Stack (cm) (h) |
|--------------------|--------------------------|
| 1                  | 7                        |
| 2                  | 9                        |
| 3                  | 11                       |
| 4                  | 13                       |
| 5                  | 15                       |

### Part a: Plot the Points from the Data Table
To plot the points, you can use the following pairs \((c, h)\):
- (1, 7)
- (2, 9)
- (3, 11)
- (4, 13)
- (5, 15)

When plotted, these points should form a straight line, showing a linear relationship between the number of cups and the height of the stack.

### Part b: What are the Variables in this Problem?
The variables are:
- Independent variable: **Number of cups (c)**.
- Dependent variable: **Height of the stack (h)**.

### Part c: What is the Relationship Between These Variables?
The relationship between the variables is **linear**. The height of the stack increases by 2 cm for every additional cup, which can be seen from the difference between the heights (7, 9, 11, 13, 15).

### Part d: Predict the Height of a Stack of Two Dozen Cups
To predict the height of a stack with 24 cups, we need to establish the pattern or equation. From the table:
- When \( c = 1 \), \( h = 7 \) cm.
- The height increases by 2 cm for each additional cup.

The equation can be expressed as:
\[
h = 2c + 5
\]
For 24 cups:
\[
h = 2(24) + 5 = 48 + 5 = 53 \text{ cm}
\]
So, the height of a stack of two dozen (24) cups is **53 cm**.

### Part e: Create an Algebraic Equation
Based on the linear relationship observed:
\[
h = 2c + 5
\]
where:
- \( h \) is the height in centimeters.
- \( c \) is the number of cups.

This equation allows you to predict the height of any stack based on the number of cups.

Would you like more details or have any questions?

Here are some follow-up questions to deepen understanding:
1. How would the equation change if the height per cup increased by 3 cm instead of 2 cm?
2. What is the height of a stack with 10 cups?
3. If the height of a stack is 23 cm, how many cups are in the stack?
4. Can you explain how the slope (2 cm per cup) is related to the physical stacking of cups?
5. What would the equation be if the first cup's height was 6 cm instead of 7 cm?

**Tip:** Always verify your equation by checking it against data points from the table before making predictions.

You have been hired by Tupper-Made Company to help create new packaging and shipping containers for its plastic cups. To ship the cups, they are stacked. The company wants to design boxes for shipping that will hold two dozen cups. The table below shows the data gathered from measuring the heights of stacks of cups. (Includes plotting points, finding variables, identifying the relationship, predicting the height of a stack of two dozen cups, and creating an algebraic equation.)

Learn how to analyze the linear relationship in stacking cups for packaging. This problem involves plotting points, understanding variables, and creating an algebraic equation to predict the height of a stack of two dozen cups. Suitable for middle school students in Grades 6-8.

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