Question

russell is participating in a fun run to raise money for the west side children’s hospital. the more laps russell runs around the track, the more money he raises for the hospital. there is a proportional relationship between the number of laps russell runs, x, and the amount of money he raises for the hospital (in dollars), y.

Explanation:

Since the problem is about a proportional relationship between the number of laps (x) and the money raised (y), we can assume it's a linear proportional relationship (direct variation), so the equation is of the form \( y = kx \), where \( k \) is the constant of proportionality. But we need to find the slope (k) from the graph. Wait, the graph has x as laps (from 1 to 10) and y as dollars (from 1 to 10). Let's assume two points on the line. Let's say when x = 10 (laps), what's y? Wait, the line seems to go from (0,0) maybe? Wait, the description says proportional relationship, so it should pass through the origin. Let's check the graph: when x=10 (laps), y=1? Wait, no, maybe the axes are labeled with x as laps (horizontal) and y as dollars (vertical). Wait, the grid: x-axis (laps) from 0 to 10, y-axis (dollars) from 0 to 10. The line is a straight line from (0,0) to maybe (10,1)? Wait, no, the green line: when x=10 (laps), y=1 dollar? Wait, maybe the slope is \( \frac{1}{10} \), so the equation is \( y = \frac{1}{10}x \), meaning for each lap, he raises 1/10 of a dollar, or 10 cents per lap. But maybe the question is to find the constant of proportionality or the equation. Let's proceed with the step-by-step:

Step1: Identify the type of relationship

The problem states a proportional relationship, so \( y = kx \), where \( k \) is the constant of proportionality (slope).

Step2: Find two points on the line

From the graph, let's assume the line passes through (0, 0) and (10, 1) (since at x=10 laps, y=1 dollar, as per the graph's arrow).

Step3: Calculate the slope (k)

The slope \( k = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 0}{10 - 0} = \frac{1}{10} = 0.1 \).

Step4: Write the equation

So the equation is \( y = 0.1x \), meaning for each lap (x), the money raised (y) is 0.1 dollars per lap, or 10 cents per lap.

Answer:

The constant of proportionality (slope) is \( \boldsymbol{\frac{1}{10}} \) (or 0.1), and the equation is \( \boldsymbol{y = \frac{1}{10}x} \) (or \( y = 0.1x \)).