To determine the correct statement regarding how the slope relates to the height of water in the pool, let's first understand what the slope of a line represents.
The slope is a measure of the rate at which one variable changes with respect to another. Specifically, in this context, the slope represents the rate of change of the height of the water with respect to time.
Given the data points in the table:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Time (min)} & \text{Height (in.)} \\
\hline 2 & 8 \\
\hline 4 & 12 \\
\hline 6 & 16 \\
\hline 8 & 20 \\
\hline 10 & 24 \\
\hline
\end{array}
\][/tex]
We find the differences in height for consecutive time intervals:
- From 2 min to 4 min: Height changes from 8 inches to 12 inches, so the difference in height is [tex]\( 12 - 8 = 4 \)[/tex] inches.
- From 4 min to 6 min: Height changes from 12 inches to 16 inches, so the difference in height is [tex]\( 16 - 12 = 4 \)[/tex] inches.
- From 6 min to 8 min: Height changes from 16 inches to 20 inches, so the difference in height is [tex]\( 20 - 16 = 4 \)[/tex] inches.
- From 8 min to 10 min: Height changes from 20 inches to 24 inches, so the difference in height is [tex]\( 24 - 20 = 4 \)[/tex] inches.
Next, we find the differences in time for these intervals:
- From 2 min to 4 min: Difference in time is [tex]\( 4 - 2 = 2 \)[/tex] minutes.
- From 4 min to 6 min: Difference in time is [tex]\( 6 - 4 = 2 \)[/tex] minutes.
- From 6 min to 8 min: Difference in time is [tex]\( 8 - 6 = 2 \)[/tex] minutes.
- From 8 min to 10 min: Difference in time is [tex]\( 10 - 8 = 2 \)[/tex] minutes.
The slope is defined as the change in height divided by the change in time:
- For the first interval: Slope = [tex]\( \frac{\Delta \text{Height}}{\Delta \text{Time}} = \frac{4 \text{ inches}}{2 \text{ minutes}} = 2 \text{ inches per minute} \)[/tex].
By calculating the slope for each interval, we see that it remains consistent at 2 inches per minute.
Therefore, the slope of 2 means that the height of the water increases by 2 inches for every minute that passes as the pool is being filled.
Hence, the correct statement that describes how the slope relates to the height of the water in the pool is:
The height of the water increases 2 inches per minute.
