Question

1. 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 total length of the row. We have two points: \((13, 23.5)\) and \((18, 31)\).

Step2: Calculate slope (additional length per cart)

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).
Substituting values: \( m=\frac{31 - 23.5}{18 - 13}=\frac{7.5}{5}=1.5 \).
This slope represents the length added by each nested cart.

Step1: Find the linear equation

Using point - slope form \( y - y_1=m(x - x_1) \). We know \( m = 1.5 \), \( x_1 = 13 \), \( y_1 = 23.5 \).
\( y-23.5 = 1.5(x - 13) \)
\( y-23.5=1.5x-19.5 \)
\( y = 1.5x + 4 \) (we can also see that the starting cart is 4 feet long, so the y - intercept is 4, which matches \( y=1.5x + 4 \) when \( x = 0 \), \( y = 4 \))

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

Substitute \( y = 43 \) into \( y=1.5x + 4 \):
\( 43=1.5x + 4 \)

Step3: Isolate \( x \)

Subtract 4 from both sides: \( 43 - 4=1.5x \)
\( 39 = 1.5x \)
Divide both sides by 1.5: \( x=\frac{39}{1.5}=26 \)
But \( x \) is the number of nested carts. The total number of carts is the number of nested carts plus 1 (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