Let's find the term that describes the slope of the line representing the data in the table by following these steps:
1. Record the data points and establish differences:
We have the following data points:
- Time (min): [tex]\[0, 1, 2, 3, 4, 5\][/tex]
- Water in Pool (gallons): [tex]\[50, 44, 38, 32, 26, 20\][/tex]
2. Determine the changes (differences):
To find the slope, we need to see how the water level changes as time progresses. We can do this by finding the differences in water levels for each consecutive time interval.
- Change in Time ([tex]\(\Delta t\)[/tex]): The time difference between consecutive measurements is constant, i.e., [tex]\(1\)[/tex] minute.
- Change in Water ([tex]\(\Delta W\)[/tex]): The change in water level each minute is calculated as follows:
[tex]\[
\Delta W_{0-1} = 44 - 50 = -6 \\
\Delta W_{1-2} = 38 - 44 = -6 \\
\Delta W_{2-3} = 32 - 38 = -6 \\
\Delta W_{3-4} = 26 - 32 = -6 \\
\Delta W_{4-5} = 20 - 26 = -6
\][/tex]
3. Calculate the slope:
The slope ([tex]\(m\)[/tex]) of a line is given by the ratio of the change in water level to the change in time:
[tex]\[
m = \frac{\Delta W}{\Delta t}
\][/tex]
Since [tex]\(\Delta W = -6\)[/tex] gallons and [tex]\(\Delta t = 1\)[/tex] minute, we have:
[tex]\[
m = \frac{-6}{1} = -6
\][/tex]
4. Describe the slope:
The negative slope indicates that the water level is decreasing over time. Hence, the slope is described as "decreasing."
In conclusion, the term that describes the slope of the line representing the volume of water in a pool over time is "decreasing".
