Given the table, we can analyze the relationship between the time (in minutes) and the height of the water (in inches) to understand how the height changes over time:
[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]
First, let's identify the pattern in the height of water given the time intervals:
- At [tex]\(2\)[/tex] minutes, the height is [tex]\(8\)[/tex] inches.
- At [tex]\(4\)[/tex] minutes, the height is [tex]\(12\)[/tex] inches.
- At [tex]\(6\)[/tex] minutes, the height is [tex]\(16\)[/tex] inches.
- At [tex]\(8\)[/tex] minutes, the height is [tex]\(20\)[/tex] inches.
- At [tex]\(10\)[/tex] minutes, the height is [tex]\(24\)[/tex] inches.
We observe that as time increases, the height of the water also increases. Let's calculate the change in height for each time interval:
1. From [tex]\(2\)[/tex] to [tex]\(4\)[/tex] minutes:
[tex]\[
\Delta \text{Height} = 12 - 8 = 4 \text{ inches}
\][/tex]
[tex]\[
\Delta \text{Time} = 4 - 2 = 2 \text{ minutes}
\][/tex]
[tex]\[
\text{Slope} = \frac{\Delta \text{Height}}{\Delta \text{Time}} = \frac{4}{2} = 2 \text{ inches per minute}
\][/tex]
2. From [tex]\(4\)[/tex] to [tex]\(6\)[/tex] minutes:
[tex]\[
\Delta \text{Height} = 16 - 12 = 4 \text{ inches}
\][/tex]
[tex]\[
\Delta \text{Time} = 6 - 4 = 2 \text{ minutes}
\][/tex]
[tex]\[
\text{Slope} = \frac{\Delta \text{Height}}{\Delta \text{Time}} = \frac{4}{2} = 2 \text{ inches per minute}
\][/tex]
3. From [tex]\(6\)[/tex] to [tex]\(8\)[/tex] minutes:
[tex]\[
\Delta \text{Height} = 20 - 16 = 4 \text{ inches}
\][/tex]
[tex]\[
\Delta \text{Time} = 8 - 6 = 2 \text{ minutes}
\][/tex]
[tex]\[
\text{Slope} = \frac{\Delta \text{Height}}{\Delta \text{Time}} = \frac{4}{2} = 2 \text{ inches per minute}
\][/tex]
4. From [tex]\(8\)[/tex] to [tex]\(10\)[/tex] minutes:
[tex]\[
\Delta \text{Height} = 24 - 20 = 4 \text{ inches}
\][/tex]
[tex]\[
\Delta \text{Time} = 10 - 8 = 2 \text{ minutes}
\][/tex]
[tex]\[
\text{Slope} = \frac{\Delta \text{Height}}{\Delta \text{Time}} = \frac{4}{2} = 2 \text{ inches per minute}
\][/tex]
The consistent slope value of [tex]\(2\)[/tex] inches per minute suggests that the rate at which the height of the water increases is constant.
Given this slope and looking at the choices:
1. The height of the water increases 2 inches per minute.
2. The height of the water decreases 2 inches per minute.
3. The height of the water was 2 inches before any water was added.
4. The height of the water will be 2 inches when the pool is filled.
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.
