To determine the slope of the line that represents the volume of water in the pool over time, we use the provided data points:
[tex]\[
\begin{tabular}{|c|c|}
\hline
Time (min) & Water in Pool (gal) \\
\hline
0 & 50 \\
\hline
1 & 44 \\
\hline
2 & 38 \\
\hline
3 & 32 \\
\hline
4 & 26 \\
\hline
5 & 20 \\
\hline
\end{tabular}
\][/tex]
We can use the formula for the slope [tex]\( m \)[/tex], which is the rate of change of the dependent variable (water in pool) with respect to the independent variable (time):
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{\text{change in water}}{\text{change in time}}
\][/tex]
To calculate the slope, we take any two points from the table. For example, let's use the first two points: [tex]\((0, 50)\)[/tex] and [tex]\((1, 44)\)[/tex].
[tex]\[
\Delta y = 44 - 50 = -6
\][/tex]
[tex]\[
\Delta x = 1 - 0 = 1
\][/tex]
Hence, the slope is:
[tex]\[
m = \frac{\Delta y}{\Delta x} = \frac{-6}{1} = -6
\][/tex]
We can validate the consistency of this slope calculation by checking other consecutive data points:
[tex]\[
\begin{aligned}
(1, 44) \text{ and } (2, 38): \quad & \Delta y = 38 - 44 = -6 \\
& \Delta x = 2 - 1 = 1 \\
& \frac{\Delta y}{\Delta x} = \frac{-6}{1} = -6
\end{aligned}
\][/tex]
[tex]\[
\begin{aligned}
(2, 38) \text{ and } (3, 32): \quad & \Delta y = 32 - 38 = -6 \\
& \Delta x = 3 - 2 = 1 \\
& \frac{\Delta y}{\Delta x} = \frac{-6}{1} = -6
\end{aligned}
\][/tex]
[tex]\[
\begin{aligned}
(3, 32) \text{ and } (4, 26): \quad & \Delta y = 26 - 32 = -6 \\
& \Delta x = 4 - 3 = 1 \\
& \frac{\Delta y}{\Delta x} = \frac{-6}{1} = -6
\end{aligned}
\][/tex]
[tex]\[
\begin{aligned}
(4, 26) \text{ and } (5, 20): \quad & \Delta y = 20 - 26 = -6 \\
& \Delta x = 5 - 4 = 1 \\
& \frac{\Delta y}{\Delta x} = \frac{-6}{1} = -6
\end{aligned}
\][/tex]
The slope remains consistent at [tex]\( -6 \)[/tex] for each pair of points.
The term that describes the slope of the line is "negative" because as time increases, the volume of water in the pool decreases, giving a negative rate of change. Therefore, the slope of the line graphed to represent the volume of water in a pool over time is negative.
