Math Problem Statement
Solution
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:
-
Define Variables:
- Let represent the number of nested carts.
- Let represent the total length of the row in feet.
-
Identify Given Points:
- When (no nested carts), feet.
- When , feet.
- When , feet.
-
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).
-
Calculate the Slope: This suggests that each additional cart adds 1.5 feet to the total length.
-
Form the Equation:
- Since the initial length (with 0 nested carts) is 4 feet, the equation can be written as:
-
Verify with the Third Point:
- Substitute into the equation:
- This confirms that the equation is correct.
-
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.
- Label the Axes:
This graph would show a linear increase in total length as the number of nested carts increases, following the equation .
Would you like further details on how to scale the graph, or do you have any other questions?
Here are five related questions to expand your understanding:
- How would the equation change if the starting cart was 5 feet instead of 4 feet?
- What would the total length be if there were 25 nested carts?
- How many nested carts are required for the row to reach 40 feet in length?
- If each cart length changed to 1.2 feet instead of 1.5 feet, how would the equation and graph change?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Equations
Slope-Intercept Form
Graphing
Formulas
y = mx + b (linear equation)
slope = (y2 - y1) / (x2 - x1)
Theorems
Linear Relationship
Suitable Grade Level
Grades 8-10
Related Recommendation
Determining the Length Each Nested Cart Adds
Find the Explicit Rule for Row Lengths of Nested Grocery Carts
Linear Relationship in Stacking Cups: Predicting Height for Tupper-Made Packaging
Understanding the Slope in a Linear Cost Graph for Baseball Hats
Linear Function to Predict Calculator Sales Based on Price