Question

the table shows the linear relationship between the amount of water in a pool and the time the pool has been draining.

time(mins): 10, 32, 68, 90
amount of water (gallons): 5,385, 5,132, 4,718, 4,465
what is the rate of change of the amount of water in the pool in gallons with respect to the number of minutes the pool has been draining?

answer; ____

Answer 1

To find the rate of change of the amount of water in the pool with respect to the number of minutes, we can calculate the slope of the linear relationship. The slope \( m \) can be determined using the formula:

\[
m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Where:
- \( y_1 \) and \( y_2 \) are the amounts of water
- \( x_1 \) and \( x_2 \) are the corresponding times

Choosing the first two points for simplicity where \( (x_1, y_1) = (10, 5385) \) and \( (x_2, y_2) = (32, 5132) \):

\[
m = \frac{5132 - 5385}{32 - 10} = \frac{-253}{22} \approx -11.5
\]

Now calculating with the second pair \( (x_1, y_1) = (32, 5132) \) and \( (x_2, y_2) = (68, 4718) \):

\[
m = \frac{4718 - 5132}{68 - 32} = \frac{-414}{36} \approx -11.5
\]

And finally with the last pair \( (x_1, y_1) = (68, 4718) \) and \( (x_2, y_2) = (90, 4465) \):

\[
m = \frac{4465 - 4718}{90 - 68} = \frac{-253}{22} \approx -11.5
\]

The rate of change of the amount of water in the pool with respect to the number of minutes the pool has been draining is approximately:

\[
\text{Rate of change} = -11.5 \text{ gallons per minute}
\]

So in your answer format:

**Answer:** -11.5