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To analyze this problem, we need to determine the relationship between the number of nested shopping carts and the total length of a row. Here’s a step-by-step breakdown:

1. **Define Variables**:
   - Let \( x \) represent the number of nested carts.
   - Let \( y \) represent the total length of the row in feet.

2. **Identify Given Points**:
   - When \( x = 0 \) (no nested carts), \( y = 4 \) feet.
   - When \( x = 13 \), \( y = 23.5 \) feet.
   - When \( x = 18 \), \( y = 31 \) feet.

3. **Formulate a Linear Equation**:
   - The relationship appears linear, as each additional cart adds a fixed length to the row. To confirm, calculate the slope (rate of increase in length per cart).

4. **Calculate the Slope**:
   \[
   \text{slope} = \frac{23.5 - 4}{13 - 0} = \frac{19.5}{13} = 1.5
   \]
   This suggests that each additional cart adds 1.5 feet to the total length.

5. **Form the Equation**:
   - Since the initial length (with 0 nested carts) is 4 feet, the equation can be written as:
   \[
   y = 1.5x + 4
   \]

6. **Verify with the Third Point**:
   - Substitute \( x = 18 \) into the equation:
     \[
     y = 1.5(18) + 4 = 27 + 4 = 31
     \]
   - This confirms that the equation is correct.

7. **Graphing the Equation**:
   - **Label the Axes**: 
     - The x-axis represents the "Number of Nested Carts" (starting from 0).
     - The y-axis represents the "Total Length in Feet."
   - **Plot Points**: Plot the points (0, 4), (13, 23.5), and (18, 31).
   - **Draw the Line**: Draw a straight line passing through these points.

This graph would show a linear increase in total length as the number of nested carts increases, following the equation \( y = 1.5x + 4 \).

Would you like further details on how to scale the graph, or do you have any other questions?

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Here are five related questions to expand your understanding:

1. How would the equation change if the starting cart was 5 feet instead of 4 feet?
2. What would the total length be if there were 25 nested carts?
3. How many nested carts are required for the row to reach 40 feet in length?
4. If each cart length changed to 1.2 feet instead of 1.5 feet, how would the equation and graph change?
5. Can we determine if the relationship is truly linear by comparing more data points?

**Tip**: In linear functions, the y-intercept represents the starting value when the independent variable (here, the number of carts) is zero.

A store is designing the space for rows of nested shopping carts. Each row has a starting cart that is 4 feet long, followed by the nested carts. (0 nested carts means there’s just the starting cart.)

• A row of 0 nested carts is 4 feet long.
• A row of 13 nested carts is 23.5 feet long.
• A row of 18 nested carts is 31 feet long.

Enter the appropriate label and scale for each axis, and drag the movable points to create a graph of the situation.

Understand how to calculate and graph the linear relationship between the number of nested shopping carts and the total row length. Given points allow calculation of the slope, and the linear equation is formed to describe the scenario. This problem is suitable for students in Grades 8-10.

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