Question

problem 5
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 (so 0 nested carts means there’s just the starting cart). the store measured a row of 13 nested carts to be 23.5 feet long, and a row of 18 nested carts to be 31 feet long.

a. create a graph of the situation.

Explanation:

Step1: Define Variables

Let \( x \) be the number of nested carts, \( y \) be the length of the row (in feet). We have two points: \((13, 23.5)\) and \((18, 31)\), plus the y - intercept when \( x = 0 \), \( y=4\) (since 0 nested carts means just the starting cart of 4 feet).

Step2: Find the Slope

The slope \( m \) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1}\). Substituting \( x_1 = 13,y_1=23.5,x_2 = 18,y_2 = 31\), we get \( m=\frac{31 - 23.5}{18 - 13}=\frac{7.5}{5}=1.5\).

Step3: Determine the Equation

Using the slope - intercept form \( y=mx + b \), we know \( b = 4\) (when \( x = 0\)) and \( m = 1.5\), so the equation is \( y=1.5x + 4\).

Step4: Plot the Points

• Plot the point \((0,4)\) (y - intercept).
• Plot the points \((13,23.5)\) and \((18,31)\).
• Draw a straight line through these points. The x - axis represents the number of nested carts (\( x \)) and the y - axis represents the length of the row (\( y \)) in feet. The scale on the x - axis can be from 0 to 20 (to cover the given values of 13 and 18) and on the y - axis from 0 to 35 (to cover the lengths 23.5 and 31 and the starting length of 4).

Answer:

To create the graph:

1. Axes Definition: Label the x - axis as "Number of Nested Carts" and the y - axis as "Length of Row (feet)".
2. Plot Points:

• Plot \((0,4)\) (since 0 nested carts give a length of 4 feet).
• Plot \((13,23.5)\) (13 nested carts, length 23.5 feet).
• Plot \((18,31)\) (18 nested carts, length 31 feet).

1. Draw Line: Draw a straight line passing through these three points. The line should follow the equation \( y = 1.5x+4\), where the slope is \( 1.5\) (representing the additional length per nested cart) and the y - intercept is 4 (length of the starting cart).