Let's address each part of the given problem step-by-step.
### Part (a): How Much Water Was in the Vat Initially?
To find out how much water was in the vat initially, we need to determine the initial amount of water before any draining occurred. Given that the water is being drained at a constant rate, we can model the situation with a linear equation of the form:
[tex]\[ \text{Water} = \text{initial amount of water} - (\text{rate of draining} \times \text{time}) \][/tex]
From the given data points:
- At 15 minutes, there are 610 liters of water.
- At 20 minutes, there are 540 liters of water.
- At 25 minutes, there are 470 liters of water.
- At 30 minutes, there are 400 liters of water.
Using the method of linear regression on these data points, we determine that the initial amount of water in the vat was approximately:
[tex]\[ \text{Initial Amount of Water} = 820 \text{ liters} \][/tex]
### Part (b): Rate of Draining
To understand the relationship between time and the amount of water in the vat, consider that as time progresses, the water level decreases. This linear relationship can be quantified by determining the slope of the line, which represents the rate of draining.
This slope (rate of draining) is calculated to be:
[tex]\[ \text{Rate of Draining} = 14 \text{ liters per minute} \][/tex]
Thus:
- As time increases by 1 minute, the amount of water in the vat decreases by 14 liters.
### Summary of Answers:
(a) The initial amount of water in the vat was [tex]\( \boxed{820} \)[/tex] liters.
(b) The relationship between time and the amount of water in the vat indicates that as time increases, the amount of water decreases at a rate of [tex]\( 14 \text{ liters per minute} \)[/tex].
