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. complete parts a) through d). 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? 5.5 hours c) how much water was in the basement before any water was pumped out? 24 thousand gallons d) after how many hours of pumping will all the water be pumped out of the basement? hours

Explanation:

Step1: Identify the x-intercept

The graph shows the volume of water \( v \) (in thousands of gallons) as a function of time \( t \) (in hours). All water is pumped out when \( v = 0 \). We look for the \( t \)-value where the line crosses the \( t \)-axis (where \( v = 0 \)).
From the graph, the line intersects the \( t \)-axis at \( t = 6 \) hours? Wait, no, wait. Wait, let's re-examine. Wait, the y-intercept is at \( t = 0 \), \( v = 24 \)? Wait, no, the graph: when \( t = 0 \), the y-value (v) is 24? Wait, no, the leftmost point is at \( t = -1 \)? Wait, no, the grid: let's check the coordinates. Wait, the line goes from (0, 24) maybe? Wait, no, the first point: when \( t = 0 \), the y-axis (v) is 24? Wait, the graph has t on the x-axis (hours) and v on the y-axis (thousands of gallons). The line passes through (0, 24) and (6, 0)? Wait, no, looking at the graph, when t = 6, v = 0? Wait, the arrow at the end of the line is at t = 6, v = 0? Wait, the x-axis is t (hours), y-axis is v (thousands of gallons). So when v = 0, t = 6? Wait, no, let's check the grid. Each square: let's see, from t = 0 to t = 6, and v from 24 to 0. Wait, the line starts at (0, 24) and goes to (6, 0). So the x-intercept (where v = 0) is at t = 6 hours? Wait, but let's confirm. Wait, the problem's part d: "After how many hours of pumping will all the water be pumped out?" That's when v = 0. So we find the t when v = 0. From the graph, the line crosses the t-axis (v = 0) at t = 6 hours? Wait, no, wait the graph: the horizontal axis (t) has marks at 0, 1, 2, 3, 4, 5, 6, 7, 8. The vertical axis (v) has marks at 0, 4, 8, 12, 16, 20, 24, 28. The line goes from (0, 24) down to (6, 0). So when v = 0, t = 6. Wait, but let's check the coordinates. Let's take two points: (0, 24) and (6, 0). The slope is \( \frac{0 - 24}{6 - 0} = -4 \). So the equation is \( v = -4t + 24 \). To find when v = 0: \( 0 = -4t + 24 \), so \( 4t = 24 \), \( t = 6 \). So the time when all water is pumped out is 6 hours.

Step2: Confirm the x-intercept

From the graph, the line intersects the t-axis (where v = 0) at t = 6 hours. So when t = 6, v = 0, meaning all water is pumped out.

Answer:

6