Question

Jace uses a cistern to fill a tub with water. When he begins, at time t=0 minutes, the tub is empty and there are 6387 L of water in the cistern. At time t=21 minutes, the cistern contains 6262 L of water and the tub contains 125 L of water. Find the average rate of change in the amount of water in the tub from t=0 to t, t=21 minutes, rounding your answer to three decimal places, with appropriate units. Then fill in the guided sentence below to interpret the meaning of your answer in context.
The average rate of change is

. It represents that the
is
at a rate of

over that time interval.

Answer 1

To find the average rate of change in the amount of water in the tub from \( t=0 \) to \( t=21 \) minutes, we need to calculate the change in the volume of water in the tub over the change in time.

1. **Initial amount of water in the tub at \( t=0 \)**: 0 L (since the tub is empty)
2. **Amount of water in the tub at \( t=21 \)**: 125 L

The change in the amount of water in the tub is:
\[
\Delta V = V(t=21) - V(t=0) = 125\, \text{L} - 0\, \text{L} = 125\, \text{L}
\]

The change in time is:
\[
\Delta t = 21\, \text{minutes} - 0\, \text{minutes} = 21\, \text{minutes}
\]

Now, we can find the average rate of change:
\[
\text{Average rate of change} = \frac{\Delta V}{\Delta t} = \frac{125\, \text{L}}{21\, \text{minutes}} \approx 5.9524 \, \text{L/min}
\]

Rounding to three decimal places, the average rate of change is approximately:
\[
5.952 \, \text{L/min}
\]

Now, let's interpret this value:

The average rate of change is **5.952 L/min**. It represents that the **amount of water in the tub** is **increasing** at a rate of **5.952 liters per minute** over that time interval.