Question

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 theres 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.
b. how much does each nested cart add to the length of the row? explain your reasoning.
c. if the store design allows for 43 feet for each row, how many total carts fit in a row?

Explanation:

Part b

Step1: Define Variables

Let \( x \) be the number of nested carts, and \( y \) be the length of the row (in feet). We have two points: \((13, 23.5)\) and \((18, 31)\).

Step2: Calculate Slope

The slope \( m \) of the line through 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 the values: \( m=\frac{31 - 23.5}{18 - 13}=\frac{7.5}{5}=1.5 \). This slope represents the length added per nested cart.

Step1: Find the Equation of the Line

Using the point - slope form \( y - y_1=m(x - x_1) \). We know \( m = 1.5 \), and we can use the point \((13,23.5)\). So \( y-23.5 = 1.5(x - 13) \). Simplify: \( y-23.5=1.5x-19.5 \), then \( y = 1.5x + 4 \). We can also check with the starting cart: when \( x = 0 \) (no nested carts), \( y=4 \), which matches the starting cart length.

Step2: Solve for \( x \) when \( y = 43 \)

Substitute \( y = 43 \) into the equation \( 43=1.5x + 4 \). Subtract 4 from both sides: \( 1.5x=43 - 4=39 \). Then \( x=\frac{39}{1.5}=26 \). But the total number of carts is the number of nested carts plus the starting cart, so total carts \(=26 + 1=27 \).

Answer:

Each nested cart adds \( 1.5 \) feet to the length of the row.

Part c