Question

comparing proportional relationships
1 the graph shows the relationship between time in seconds and volume of water in milliliters for two graduated cylinders.

a. what is the unit rate of the proportional relationship represented by the red line? how do you know?

b. what is the unit rate of the proportional relationship represented by the blue line? how do you know?

Explanation:

Part a

Step1: Identify points on red line

From the graph, red line passes through (10, 90) (assuming red is the slower one? Wait, no, let's check. Wait, maybe red line: at x=10, y=90? Wait, no, maybe the red line (the less steep one) at x=10, y=90? Wait, no, let's see. Wait, unit rate is slope, which is y/x for proportional (passes through origin). So for red line, let's take x=10, y=90? Wait, no, maybe the blue line is steeper. Wait, maybe I mixed up. Wait, the problem says red line. Let's assume red line: when time (x) is 10 seconds, volume (y) is 90 milliliters? Wait, no, maybe at x=5, y=45? Wait, no, let's do it properly. Unit rate for proportional relationship is (volume)/(time) since it's volume over time. So for red line, let's pick a point. Let's say at x=10, y=90. Then unit rate is 90/10 = 9 mL per second? Wait, no, wait the graph: y-axis is volume (mL), x-axis is time (seconds). So for red line (the one with lower slope), at x=10, y=90? Wait, no, maybe the blue line is at x=5, y=90? Wait, the graph has y from 0 to 100, x from 0 to 10. Let's re-express. Let's take the red line: when x=10, y=90? Then unit rate is 90/10 = 9 mL/s? Wait, no, maybe I got the lines wrong. Wait, the problem says "red line" and "blue line". Let's assume the red line (the less steep) has a point (10, 90), so unit rate is 90/10 = 9 mL per second. Wait, no, maybe at x=5, y=45? Then 45/5=9. Or x=10, y=90: 90/10=9. So unit rate is 9 mL per second. Because unit rate is the slope, which is y/x for proportional (since it passes through (0,0)). So for red line, take a point (x,y) on it, then unit rate = y/x.

Step2: Calculate unit rate

Take x=10, y=90 (assuming red line at x=10 is 90). Then unit rate = 90 / 10 = 9 mL per second.

Step1: Identify points on blue line

For the blue line (steeper), let's take a point. At x=5, y=90? Wait, no, at x=5, y=90? Then 90/5=18? Wait, no, maybe at x=5, y=90? Wait, the graph: if blue line is steeper, at x=5, y=90? Then unit rate is 90/5=18. Or x=10, y=180? But y-axis only goes to 100? Wait, maybe the y-axis is up to 100, so maybe at x=5, y=90? Wait, the original graph: the blue line (steeper) at x=5 is at y=90? Then 90/5=18. Wait, no, maybe the y-axis is volume in milliliters, x-axis time in seconds. Let's check: if blue line passes through (5, 90), then unit rate is 90/5=18 mL per second. Or (10, 180) but y-axis is up to 100? Wait, maybe the graph's y-axis is labeled up to 100, but maybe the numbers are 10,20,...100. Wait, the y-axis has 10,20,30,40,50,60,70,80,90,100? Wait, the first mark after 0 is 10, then 20, etc. So at x=5, blue line is at y=90? Then 90/5=18. So unit rate is 18 mL per second. Because for proportional, unit rate is y/x. So take x=5, y=90: 90/5=18.

Step2: Calculate unit rate

Take x=5, y=90 (on blue line). Unit rate = 90 / 5 = 18 mL per second.

Answer:

(part a):
The unit rate of the red line is 9 milliliters per second. We know this because for a proportional relationship (passing through the origin), the unit rate is the slope, calculated as \( \frac{\text{volume}}{\text{time}} \). Taking the point (10, 90) on the red line, \( \frac{90}{10} = 9 \) mL per second.

Part b