Solution for the given problem:

(a) How much water was in the vat when the workers started draining it?

From the table, observe the pattern in water levels at specific times. To find the initial amount of water (before time = 15 minutes), determine the rate of change (draining rate), then use it to calculate the starting amount.

The rate of change can be calculated using two data points:

• Between 15 minutes (560 liters) and 20 minutes (480 liters), the change in water is: $\text{Rate of change} = \frac{\text{Change in water}}{\text{Change in time}} = \frac{560 - 480}{20 - 15} = \frac{80}{5} = 16 \text{ liters per minute}.$

Using this rate of change, extrapolate backward to time = 0: $\text{Initial water} = \text{Water at } t=15 + (\text{Rate of change} \times 15) = 560 + (16 \times 15) = 560 + 240 = 800 \text{ liters}.$

Answer (a): 800 liters.

(b) Describe the relationship and find the rate:

The water level is decreasing as time increases. The constant rate of decrease is 16 liters per minute, as calculated above.

Answer (b):

• Statement: "As time increases, the amount of water in the vat decreases."
• Rate of decrease: 16 liters per minute.

Let me know if you'd like further clarification or have any additional questions!

Additional Questions:

1. How would the calculation change if the rate of change varied over time?
2. What is the water level at 35 minutes if the draining continues at the same rate?
3. How much total time would it take to completely drain the vat?
4. Can you plot the water level versus time for better visualization?
5. How could this data help in optimizing the drainage process?

Tip: Always double-check the rate of change when working with constant patterns to ensure accuracy!