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.

chart: x-axis (t) labeled hours from -1 to 8, y-axis (v) labeled thousands of gallons from -4 to 28. blue line with arrows, starting at (0,24), ending at (6,0), passing through (3,12) etc. zoom in/out/expand buttons.

complete parts a) through d).

a) how much water is in the basement after 3 hours of pumping?
box thousand gallons

Explanation:

Step1: Identify the linear model

The graph is a linear plot of volume \( v \) (in thousands of gallons) vs. time \( t \) (in hours). We can find the equation of the line. From the graph, when \( t = 0 \), \( v = 28 \) (y-intercept). The slope \( m \) can be calculated as \( \frac{\Delta v}{\Delta t} \). For example, from \( t = 0 \) to \( t = 6 \), \( v \) goes from 28 to 0, so \( m=\frac{0 - 28}{6 - 0}=-\frac{14}{3}\). Wait, alternatively, looking at the grid, at \( t = 3 \), we can check the value. Wait, maybe easier to read the graph: at \( t = 3 \) hours, what's \( v \)?

Looking at the graph, the x-axis is hours (t), y-axis is thousands of gallons (v). Let's check the coordinates. At \( t = 0 \), \( v = 28 \). At \( t = 3 \), let's see the grid. Each grid square: x-axis, from 0 to 6 is 6 units, y-axis from 0 to 28 is 28 units. Wait, the line passes through (0,28) and (6,0). So the equation is \( v = -\frac{28}{6}t + 28=-\frac{14}{3}t + 28 \).

Step2: Substitute \( t = 3 \)

Substitute \( t = 3 \) into the equation: \( v = -\frac{14}{3}(3)+28=-14 + 28 = 14 \). Alternatively, by looking at the graph, at \( t = 3 \) hours, the y-value (v) is 14 (since from t=0 (28) to t=6 (0), at t=3, it's halfway? Wait, 28 to 0 over 6 hours, so at 3 hours, it's 28 - 14 = 14. So the volume at t=3 is 14 thousand gallons.

Answer:

14