Question

the graph shows the amount of water in the pool vs. time relationship for filling a kiddie pool with a small hose and a large hose.
select all of the true statements.

• the amount of water in the pool vs. time relationship for both the large and small hoses are proportional relationships.
• the relationship between time, t, and the amount of water, w, for the small hose can be represented by the equation w = 2.5t.
• it takes 6 minutes to fill the pool with the large hose and 12 minutes with the small hose.
• if the pool is filled with the small hose for 4 minutes it will have 10 gallons of water.
• the speed as a unit rate for filling the pool with the large hose is 2.5 gallons per minute.

Explanation:

Step1: Analyze Proportional Relationship

Proportional relationships have graphs passing through the origin (\( (0,0) \)) and are linear. Both lines start at \( (0,0) \), so they are proportional. So the first statement is true.

Step2: Check Small Hose Equation

For the small hose, let's find the slope (unit rate). From the graph, at \( t = 4 \) min, \( w = 4 \) gal? Wait, no, wait. Wait, the small hose line: let's take two points. At \( t = 4 \), what's \( w \)? Wait, no, looking at the graph, the small hose: when \( t = 4 \), \( w = 4 \)? Wait, no, maybe I misread. Wait, the large hose: at \( t = 4 \), \( w = 10 \)? Wait, no, the graph: y-axis is water (gal), x-axis time (min). Let's re-examine. The small hose line: let's take \( t = 4 \), \( w = 4 \)? No, wait, the blue line (small hose) at \( t = 4 \), \( w = 4 \)? Wait, no, maybe the large hose: at \( t = 4 \), \( w = 10 \)? Wait, no, the first statement: both are proportional (pass through origin, linear). So first statement: true.

Second statement: small hose equation \( w = 2.5t \). Let's check. If \( t = 4 \), \( w = 2.5*4 = 10 \)? But the small hose at \( t = 4 \), looking at the graph, is it 10? No, the small hose line (blue) at \( t = 4 \), seems to be at \( w = 4 \)? Wait, no, maybe I messed up. Wait, the large hose (green) is steeper. Let's calculate the slope of small hose. Let's take two points on small hose: (0,0) and (12, 15)? Wait, no, the graph's x-axis goes to 16, y-axis to 16. Wait, the small hose line: at \( t = 12 \), \( w = 15 \)? No, the blue line at \( t = 12 \), \( w = 15 \)? Wait, no, the problem says "small hose" equation \( w = 2.5t \). Let's check \( t = 4 \): \( 2.5*4 = 10 \). If the small hose at \( t = 4 \) has \( w = 10 \), then yes. Wait, maybe the graph's grid: each square is 2 units? Wait, the y-axis: 0,4,8,12,16. x-axis: 0,4,8,12,16. So at \( t = 4 \), small hose: let's see, the blue line at \( t = 4 \), is it at \( w = 10 \)? Wait, the large hose (green) at \( t = 4 \) is at \( w = 10 \)? No, the green line (large hose) at \( t = 4 \) is at \( w = 10 \), and blue (small hose) at \( t = 4 \) is at \( w = 4 \)? No, that can't be. Wait, maybe the small hose's slope: let's take \( t = 12 \), \( w = 15 \)? Then slope is \( 15/12 = 1.25 \). But \( 2.5 \) is 5/2. Wait, maybe the large hose: slope is \( 10/4 = 2.5 \). Oh! Wait, I mixed up the hoses. The large hose (green) has steeper slope. So the large hose: at \( t = 4 \), \( w = 10 \), so slope \( 10/4 = 2.5 \), so large hose equation \( w = 2.5t \). But the second statement says small hose equation \( w = 2.5t \). So that's false. So second statement: false.

Third statement: takes 6 min to fill with large hose and 12 min with small hose. Let's see: large hose: when does \( w = 15 \) (assuming pool capacity is 15 gal? Wait, the graph's y-axis goes to 16. Wait, maybe the pool capacity is 15 gal? Wait, large hose: rate is 2.5 gal/min (from \( w = 2.5t \)). So time to fill: \( t = 15 / 2.5 = 6 \) min. Small hose: let's find its rate. If small hose at \( t = 12 \), \( w = 15 \)? Wait, no, the blue line (small hose) at \( t = 12 \), \( w = 15 \)? Then rate is \( 15/12 = 1.25 \) gal/min. Then time to fill 15 gal: \( 15 / 1.25 = 12 \) min. So third statement: true.

Fourth statement: small hose for 4 min, 10 gal? Wait, no, small hose rate is 1.25 gal/min. So 4 min: \( 1.25*4 = 5 \) gal? Wait, no, the fourth statement: "If the pool is filled with the small hose for 4 minutes it will have 10 gallons of water." But small hose rate is 1.25, so 4*1.25 = 5. So fourth statement: false.

Fifth statement: large hose speed (unit rate) is 2.5 gal/min. From large hose: at \( t = 4 \), \( w = 10 \) (since \( 2.5*4 = 10 \)). So rate is \( 10/4 = 2.5 \) gal/min. So fifth statement: true.

Wait, let's re-express:

1. Both proportional: true (pass through origin, linear).

2. Small hose equation \( w = 2.5t \): false (small hose rate is lower, large hose is 2.5).

3. 6 min (large) and 12 min (small): true (large: 15 gal / 2.5 gal/min = 6 min; small: 15 gal / 1.25 gal/min = 12 min).

4. Small hose 4 min: 10 gal: false (1.25*4=5).

5. Large hose speed 2.5 gal/min: true (10 gal / 4 min = 2.5).

Wait, but maybe the pool capacity is 15 gal? Let's check: large hose: 2.5 gal/min * 6 min = 15 gal. Small hose: 1.25 gal/min * 12 min = 15 gal. So third statement: true.

So true statements: first, third, fifth? Wait, no, let's recheck:

First statement: both proportional (pass through (0,0), linear) → true.

Second: small hose \( w = 2.5t \) → large hose has rate 2.5, small hose lower → false.

Third: 6 min (large) and 12 min (small) → true (15 / 2.5 = 6; 15 / 1.25 = 12).

Fourth: small hose 4 min → 10 gal → 1.25*4=5 → false.

Fifth: large hose speed 2.5 gal/min → true (10/4=2.5).

Wait, but maybe the graph's y-axis at t=4, large hose is at 10 gal (so 10/4=2.5). Small hose at t=4 is at 5 gal (5/4=1.25). So:

1. Both proportional: true (lines through origin, linear).

3. 6 min (large: 15/2.5=6) and 12 min (small:15/1.25=12): true.

5. Large hose speed 2.5 gal/min: true.

Wait, but the options:

First option: "The amount of water in the pool vs. time relationship for both the large and small hoses are proportional relationships." → true.

Third option: "It takes 6 minutes to fill the pool with the large hose and 12 minutes with the small hose." → true.

Fifth option: "The speed as a unit rate for filling the pool with the large hose is 2.5 gallons per minute." → true.

Wait, but maybe I made a mistake with the small hose equation. Let's check the second option again: "The relationship between time, \( t \), and the amount of water, \( w \), for the small hose can be represented by the equation \( w = 2.5t \)." If small hose rate is 1.25, then \( w = 1.25t \), so second option is false.

So true statements: first, third, fifth? Wait, but let's check the graph again. The large hose (green) at t=4, w=10 (so 10/4=2.5). Small hose (blue) at t=4, w=5 (5/4=1.25). So:

1. Both proportional: true (pass through (0,0), linear).

3. Time to fill: large hose: 15 gal / 2.5 gal/min = 6 min; small hose:15 gal / 1.25 gal/min=12 min → true.

5. Large hose rate: 2.5 gal/min → true.

So the true statements are:

- The amount of water in the pool vs. time relationship for both the large and small hoses are proportional relationships.

- It takes 6 minutes to fill the pool with the large hose and 12 minutes with the small hose.

- The speed as a unit rate for filling the pool with the large hose is 2.5 gallons per minute.

Wait, but let's confirm the third statement: if the pool capacity is 15 gal (since large hose at 6 min: 2.5*6=15; small hose at 12 min:1.25*12=15). So yes.

So the true statements are the first, third, and fifth? Wait, but the options:

1. First option: checkbox.

2. Second: checkbox.

3. Third: checkbox.

4. Fourth: checkbox.

5. Fifth: checkbox.

So the true ones are:

1. The amount of water in the pool vs. time relationship for both the large and small hoses are proportional relationships.

3. It takes 6 minutes to fill the pool with the large hose and 12 minutes with the small hose.

5. The speed as a unit rate for filling the pool with the large hose is 2.5 gallons per minute.

Wait, but maybe the second option is true? Wait, no, small hose rate is 1.25, so \( w = 1.25t \), not 2.5t. So second option is false.

Fourth option: small hose 4 min → 10 gal. 1.25*4=5, so false.

So the true statements are the first, third, and fifth.

Answer:

• The amount of water in the pool vs. time relationship for both the large and small hoses are proportional relationships.
• It takes 6 minutes to fill the pool with the large hose and 12 minutes with the small hose.
• The speed as a unit rate for filling the pool with the large hose is 2.5 gallons per minute.