To find the rate of change of the water being drained from the bathtub, we need to determine how the amount of water (in gallons) changes with respect to time (in minutes). This can be done by calculating the average rate of change over the given time intervals.
The data from the table is as follows:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Time (min)} & \text{Water (gal)} \\
\hline
7 & 38.5 \\
\hline
8 & 36.25 \\
\hline
9 & 34 \\
\hline
10 & 31.75 \\
\hline
\end{array}
\][/tex]
### Step-by-Step Solution:
1. Identify the intervals:
The intervals between the time measurements are:
- From 7 to 8 minutes
- From 8 to 9 minutes
- From 9 to 10 minutes
2. Calculate the rates of change for each interval:
The rate of change for each interval is calculated as:
[tex]\[
\text{Rate of change} = \frac{\text{Change in water}}{\text{Change in time}}
\][/tex]
- Interval (7 to 8 minutes):
[tex]\[
\text{Rate of change} = \frac{36.25 - 38.5}{8 - 7} = \frac{-2.25}{1} = -2.25 \text{ gallons per minute}
\][/tex]
- Interval (8 to 9 minutes):
[tex]\[
\text{Rate of change} = \frac{34 - 36.25}{9 - 8} = \frac{-2.25}{1} = -2.25 \text{ gallons per minute}
\][/tex]
- Interval (9 to 10 minutes):
[tex]\[
\text{Rate of change} = \frac{31.75 - 34}{10 - 9} = \frac{-2.25}{1} = -2.25 \text{ gallons per minute}
\][/tex]
3. Compute the average rate of change:
Since each interval has the same rate of change:
[tex]\[
\text{Average rate of change} = \frac{\sum \text{rates of change}}{\text{number of intervals}}
\][/tex]
There are 3 intervals and each interval's rate of change is -2.25 gallons per minute, so:
[tex]\[
\text{Average rate of change} = \frac{-2.25 + (-2.25) + (-2.25)}{3} = \frac{-6.75}{3} = -2.25 \text{ gallons per minute}
\][/tex]
Therefore, the rate of change for the relationship between time and the amount of water in the bathtub is [tex]\(-2.25\)[/tex] gallons per minute.
