To find the term that describes the slope of the line representing the volume of water in the pool over time, we first need to understand the concept of slope. The slope of a line in a graph is a measure of how steep the line is. It is calculated as the change in the [tex]\(y\)[/tex]-values divided by the change in the [tex]\(x\)[/tex]-values. In this context, the [tex]\(y\)[/tex]-values represent the volume of water in gallons, and the [tex]\(x\)[/tex]-values represent the time in minutes.
Let's break down the process step-by-step using the data provided:
1. Determine the Change in Water Volume:
- The initial water volume (at [tex]\(t = 0\)[/tex] minutes) is 50 gallons.
- The final water volume (at [tex]\(t = 5\)[/tex] minutes) is 20 gallons.
- The change in water volume ([tex]\(\Delta y\)[/tex]) is calculated as:
[tex]\[
\Delta y = \text{initial volume} - \text{final volume} = 50 - 20 = 30 \text{ gallons}
\][/tex]
2. Determine the Change in Time:
- The initial time is 0 minutes.
- The final time is 5 minutes.
- The change in time ([tex]\(\Delta x\)[/tex]) is calculated as:
[tex]\[
\Delta x = \text{final time} - \text{initial time} = 5 - 0 = 5 \text{ minutes}
\][/tex]
3. Calculate the Slope:
- The slope ([tex]\(m\)[/tex]) is given by the ratio of the change in water volume to the change in time:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{30 \text{ gallons}}{5 \text{ minutes}} = 6.0 \text{ gallons per minute}
\][/tex]
Now that we have calculated the slope, we need to describe it in context. The slope of [tex]\(-6.0\)[/tex] gallons per minute indicates that for each minute that passes, the volume of water in the pool decreases by 6 gallons. This negative slope tells us the rate at which the water level is dropping over time.
Therefore, the term that best describes the slope of the line of a graph representing the data in the table is:
"The slope of the line is -6 gallons per minute." This indicates that the volume of water in the pool decreases by 6 gallons every minute.
