Problem 71 - Algebra and Trigonometry

Question

Torricelli's Law A tank holds 50 gallons of water, which drains from a leak at the bottom, causing the tank to empty in 20 minutes. The tank drains faster when it is nearly full because the pressure on the leak is greater. Torricelli's Law gives the volume of water remaining in the tank after \(t\) minutes as $$V(t)=50\left(1-\frac{t}{20}\right)^{2} \quad 0 \leq t \leq 20$$ (a) Find \(V(0)\) and \(V(20)\) . (b) What do your answers to part (a) represent? (c) Make a table of values of \(V(t)\) for \(t=0,5,10,15,20\)

Step-by-Step Solution

Answer

(a) \(V(0)=50\), \(V(20)=0\); (b) Initial and empty volumes; (c) Table: 50, 28.125, 12.5, 3.125, 0.

Step 1: Calculate V(0)

Substitute \(t = 0\) into the formula \(V(t) = 50\left(1-\frac{t}{20}\right)^{2}\). This gives us \(V(0) = 50\left(1-\frac{0}{20}\right)^{2} = 50\left(1^2\right) = 50\).

Step 2: Calculate V(20)

Substitute \(t = 20\) into the formula \(V(t) = 50\left(1-\frac{t}{20}\right)^{2}\). This gives us \(V(20) = 50\left(1-\frac{20}{20}\right)^{2} = 50(0^2) = 0\).

Step 3: Interpret V(0) and V(20)

The value \(V(0) = 50\) represents the volume of water in the tank at the start (0 minutes), which is the initial volume of water. The value \(V(20) = 0\) represents the volume of water in the tank at 20 minutes, which indicates the tank is empty at this time.

Step 4: Formulate Table of Values for V(t)

Calculate \(V(t)\) for \(t = 0, 5, 10, 15, 20\):- For \(t = 0\), \(V(0) = 50\).- For \(t = 5\), \(V(5) = 50\left(1-\frac{5}{20}\right)^{2} = 50\left(0.75\right)^{2} = 50(0.5625) = 28.125\).- For \(t = 10\), \(V(10) = 50\left(1-\frac{10}{20}\right)^{2} = 50(0.5^2) = 50(0.25) = 12.5\).- For \(t = 15\), \(V(15) = 50\left(1-\frac{15}{20}\right)^{2} = 50(0.25^2) = 50(0.0625) = 3.125\).- For \(t = 20\), \(V(20) = 0\).Thus, the table is:\[\begin{array}{c|c} t & V(t) \\hline 0 & 50 \ 5 & 28.125 \ 10 & 12.5 \ 15 & 3.125 \ 20 & 0 \ \end{array}\]

Key Concepts

Volume of Water

Understanding the volume of water in a tank is crucial to solving the problem of how quickly or slowly it drains. At the start, the tank holds a substantial 50 gallons of water. This is the maximum volume the tank can hold at time zero, represented mathematically as \( V(0) = 50 \). The volume decreases over time until it eventually reaches zero when the tank is empty.
The formula provided by Torricelli's Law, \( V(t) = 50\left(1-\frac{t}{20}\right)^2 \), captures this behavior over time, showing how the volume changes as \( t \) ranges from 0 to 20 minutes. This expression accurately models the decrease in volume, accounting for the reduced water pressure as the tank empties.

Draining Tank Problem

The draining tank problem is a classical physics challenge often encountered in scenarios involving fluid dynamics. The tank begins with a full 50 gallons of water and drains completely in 20 minutes due to a leak at the bottom. Torricelli's Law provides a practical way to represent the volume of water remaining in the tank at any given time.
This problem can be visualized by considering the rate at which water exits the tank. Initially, the tank drains rapidly, but as the water level lowers, the rate decreases due to reduced water pressure. The core of solving a draining tank problem lies in understanding how pressure influences the flow rate.

Pressure Effect on Flow

Pressure is a significant factor in determining how fast water flows from a tank. In the context of our problem, as the water level decreases, the pressure also reduces, leading to a slower flow rate.
Initially, the pressure is highest, pushing water out more swiftly through the leak. As time progresses, the pressure exerted by the remaining water lowers. Consequently, Torricelli's Law reflects this by showing a rapid decrease in volume at the start, which slows as time goes on. This pressure-dependent flow is crucial to understanding how tanks behave when draining.

Time-Dependent Function

The equation \( V(t) = 50\left(1-\frac{t}{20}\right)^2 \) is a time-dependent function representing the volume of water left in the tank as time passes. It's crucial to understand how such functions work because they relate different variables—in this case, time and volume.
The formula is structured to show that time \( t \) directly affects the volume of remaining water. At \( t = 0 \), you have the initial volume; at \( t = 20 \), the volume drops to zero. By plugging various values of \( t \), such as 0, 5, 10, 15, and 20 minutes, into the formula, we can determine the exact amount of water left in the tank at those times. This enables us to make predictions about the tank’s behavior at any point during the draining process.