Question

water is steadily pumped out of a flooded basement. let v be the volume of water (in thousands of gallons) that remains in the basement t hours after the water began to be pumped. a linear model is shown below.
a) how much water is in the basement after 3 hours of pumping?
12 thousand gallons
b) after how many hours of pumping will 2 thousand gallons remain in the basement?
□ hours

Explanation:

Step1: Find the linear equation

The graph is a line. Let's find the slope \( m \) and the y-intercept \( b \). From the graph, when \( t = 0 \), \( v = 24 \) (wait, no, the y-intercept seems to be 24? Wait, the first point: when \( t = 0 \), \( v = 24 \)? Wait, no, looking at the grid, when \( t = 0 \), the value is 24? Wait, the line passes through (0, 24) and (6, 0). So slope \( m=\frac{0 - 24}{6 - 0}=\frac{-24}{6}=-4 \). So the equation is \( v=-4t + 24 \).

Step2: Solve for t when v = 2

We set \( v = 2 \) in the equation \( 2=-4t + 24 \). Subtract 24 from both sides: \( 2-24=-4t \) → \( -22=-4t \)? Wait, no, wait maybe I made a mistake in the y-intercept. Wait, looking at the graph again: when \( t = 0 \), the y-value is 28? Wait, the first point is (0, 28)? Wait, the grid: the y-axis starts at -1, and the first point is at (0, 28)? Wait, no, the blue line starts at (0, 28)? Wait, no, the graph has a point at (0, 24)? Wait, maybe I misread. Wait, let's check the points. When \( t = 0 \), the value is 24? Wait, when \( t = 3 \), \( v = 12 \) (from part a). Let's use two points: (3, 12) and (0, 24). Then slope \( m=\frac{12 - 24}{3 - 0}=\frac{-12}{3}=-4 \). So equation is \( v=-4t + 24 \). Wait, when \( t = 0 \), \( v = 24 \). When \( t = 3 \), \( v=-12 + 24 = 12 \), which matches part a. So the equation is correct. Now, for part b, set \( v = 2 \):

\( 2=-4t + 24 \)

Subtract 24 from both sides: \( 2 - 24=-4t \) → \( -22=-4t \)? Wait, that can't be. Wait, maybe the y-intercept is 28? Wait, let's check the graph again. The top of the y-axis is 28. Let's take (0, 28) and (6, 0). Then slope \( m=\frac{0 - 28}{6 - 0}=\frac{-28}{6}=-\frac{14}{3}\)? No, but part a says at t=3, v=12. Let's check with (3,12) and (0,28): slope is \( \frac{12 - 28}{3 - 0}=\frac{-16}{3}\), which doesn't match. Wait, maybe the correct points are (0,24) and (6,0), and part a: at t=3, v=12. Let's check: \( v=-4(3)+24 = -12 +24=12 \), which matches. So equation is \( v=-4t +24 \). Now, set \( v=2 \):

\( 2=-4t +24 \)

Subtract 24: \( -22=-4t \) → \( t=\frac{22}{4}=5.5 \)? Wait, that's not right. Wait, maybe the equation is \( v=-4t +28 \)? Let's check: when t=0, v=28. When t=3, v=-12 +28=16, but part a says 12. So that's wrong. Wait, part a says after 3 hours, 12 thousand gallons. So (3,12) is a point. Let's find the equation using (3,12) and (6,0). Slope \( m=\frac{0 - 12}{6 - 3}=\frac{-12}{3}=-4 \). So equation: \( v - 12=-4(t - 3) \) → \( v - 12=-4t +12 \) → \( v=-4t +24 \). So that's correct. Now, when v=2:

\( 2=-4t +24 \)

\( -4t=2 -24=-22 \)

\( t=\frac{22}{4}=5.5 \)? But that's 5.5 hours. Wait, but maybe I made a mistake. Wait, let's check the graph again. The line goes from (0,24) to (6,0). So when t=6, v=0. So the equation is correct. Wait, but maybe the y-intercept is 28. Let's see: if (0,28) and (3,12), then slope is (12-28)/3= -16/3. Then equation is v= -16/3 t +28. When t=3, v= -16 +28=12, which matches part a. Ah! So I made a mistake in the y-intercept. The correct y-intercept is 28. So let's redo the equation.

Points: (0,28) and (3,12). Slope \( m=\frac{12 - 28}{3 - 0}=\frac{-16}{3}\)? No, wait (3,12) and (6,0): slope is (0-12)/(6-3)= -12/3= -4. Wait, (3,12) and (6,0): slope -4. So (0,24) is not on the line. Wait, (0,28), (3,12), (6,0): let's check the differences. From t=0 to t=3: t increases by 3, v decreases by 16 (28 to 12). From t=3 to t=6: t increases by 3, v decreases by 12 (12 to 0). No, that's not linear. Wait, the graph is a straight line, so the slope must be constant. So (0,28), (3,12), (6,0): the slope from (0,28) to (3,12) is (12-28)/3= -16/3 ≈ -5.333. From (3,12) to (6,0) is (0-12)/3= -4. So that's not linear. Therefore, my initial mistake: the correct points are (0,24), (3,12), (6,0). So slope is -4. So equation is v= -4t +24. Then when v=2:

2 = -4t +24 → -4t = 2 -24 = -22 → t= 22/4=5.5. But that's 5.5 hours. Wait, but maybe the graph has a different y-intercept. Wait, the problem says "a linear model is shown below". Let's look at the graph again. The y-axis is "Thousands of gallons" and the x-axis is "Hours". The line starts at (0,24) (since at t=0, it's 24 thousand gallons) and goes to (6,0). So the equation is v= -4t +24. So when v=2:

2 = -4t +24 → -4t = -22 → t= 5.5. But maybe the answer is 6.5? Wait, no, let's check the grid. Each square is 1 unit? Let's see, when t=0, v=24; t=1, v=20; t=2, v=16; t=3, v=12; t=4, v=8; t=5, v=4; t=6, v=0. Wait, that's a slope of -4. So at t=0, 24; t=1,20; t=2,16; t=3,12; t=4,8; t=5,4; t=6,0. So the equation is v= -4t +24. So when v=2, we have 2= -4t +24 → -4t= -22 → t=5.5. But the problem is asking for when 2 thousand gallons remain. So t=5.5 hours. Wait, but maybe I made a mistake in the y-intercept. Wait, the first point is (0,28)? Let's check the graph again. The top of the y-axis is 28, so maybe (0,28). Then t=0, v=28; t=1, v=24; t=2, v=20; t=3, v=16; t=4, v=12; t=5, v=8; t=6, v=4; t=7, v=0. Ah! That makes sense. So the slope is -4, and y-intercept 28. So equation is v= -4t +28. Let's check: when t=3, v= -12 +28=16? No, part a says 12. Wait, part a says after 3 hours, 12 thousand gallons. So (3,12) must be on the line. So if (3,12) is on the line, then 12= -4(3) +28 → 12= -12 +28=16. No, that's not. Wait, maybe the slope is - (24 - 12)/3= -4. Wait, 24 to 12 in 3 hours, so 12 gallons per 3 hours, so 4 gallons per hour. So the rate is 4 thousand gallons per hour. So initial volume is 24 thousand gallons? No, part a says after 3 hours, 12. So 24 - 12=12 gallons pumped in 3 hours, so 4 gallons per hour. So initial volume is 24. So equation v=24 -4t. Then when v=2, 2=24 -4t → 4t=22 → t=5.5. But the graph shows that at t=6, v=0. So maybe the initial volume is 24, and it takes 6 hours to pump out 24 gallons (4 per hour). So when v=2, t=(24 -2)/4=22/4=5.5. So the answer is 5.5 hours.

Answer:

5.5