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Understanding the volume of water in a tank is crucial when dealing with Torricelli's Law as it helps us quantify the amount of water available at any given moment. In this specific exercise, we're given a formula that determines the volume of water remaining in the tank over time. The equation for this is \[ V(t) = 50\left(1 - \frac{t}{20}\right)^2 \]where \( V(t) \) represents the volume of water in gallons at time \( t \) in minutes. When we calculate \( V(0) \), it equals 50 gallons, indicating the tank's full capacity. At \( V(20) \), the volume is reduced to 0 gallons, showing the tank is completely empty. These calculations help us visualize the water level dropping as time progresses.

The draining process is a core aspect of this exercise, demonstrating how the volume reduces as time passes. The given formula models how water pours out of a tank due to a leak. Initially, the water flows quickly because of the higher water pressure exerted by the full tank on the leak.To see this process, we need to examine different points in time:

• At \( t = 0 \), the tank is at full capacity with 50 gallons.
• By \( t = 10 \), halfway through the draining period, the volume drops significantly to 12.5 gallons.
• Finally, at \( t = 20 \), it's completely drained.

This step-by-step decrease in volume exemplifies how the draining speed correlates with the amount of water left, aligning with Torricelli's principle.

Pressure plays a pivotal role in understanding the draining process. When a tank is nearly full, the weight of the water exerts more pressure on the leak, causing water to drain faster. Initially, we observe a rapid decrease in volume due to this pressure effect.Torricelli's Law illustrates that:

• As the volume decreases, the pressure also reduces.
• This results in a slower draining rate as time progresses.

Therefore, the changing pressure throughout this process accounts for the non-linear water level drop over the 20-minute duration. This principle explains why the tank empties entirely only at \( t = 20 \), despite the rapid initial outflow.

Time calculation in this context is about predicting how long it takes for water to drain under specific conditions. By applying Torricelli's Law, we can identify key points at which varying amounts of water remain.Using the formula:

• Determine the remaining volume at given \( t \) values such as 5, 10, 15 minutes.
• Observe the pattern or curve that develops, confirming the time when the tank becomes empty.

This predictive capability is useful for knowing exactly when a tank will become empty. It also serves practical purposes, aiding in planning and resource management when dealing with fluid containers. Understanding time calculation in relation to the draining process ensures accurate assessments and problem-solving related to real-life scenarios.