Problem 50 - Calculus Single Variable

Question

A cubic tank filled with water has side length \(1 \mathrm{~m}\). At time \(t=0\), a circular hole of radius \(4 \mathrm{~cm}\) opens up on the bottom of the tank. If the proportionality constant \(k\) in Torricelli's Law is given by \(k=5 / \pi \mathrm{m}^{1 / 2} / \mathrm{s},\) how long does it take the tank to drain?

Step-by-Step Solution

Answer

The tank takes approximately 227 seconds to drain.

Step 1: Understand Torricelli's Law

Torricelli's Law states that the speed, \(v\), of a liquid flowing out of an orifice under the force of gravity is \(v = \sqrt{2gh}\), where \(g\) is the acceleration due to gravity and \(h\) is the height of the liquid above the hole. The outflow rate, \(Q\), from the orifice, is given by \(Q = A \cdot v = A \cdot \sqrt{2gh}\), where \(A\) is the area of the hole.

Step 2: Determine the Area of the Hole

The radius of the hole is given as \(4 \text{ cm} = 0.04 \text{ m}\). Convert to meters since SI units are needed. The area of the hole, \(A\), is given by \(A = \pi r^2\). Thus, \(A = \pi (0.04)^2 = 0.0016\pi \text{ m}^2\).

Step 3: Set Up the Differential Equation

The rate at which the volume, \(V\), of water changes is related to the outflow rate: \(\frac{dV}{dt} = -A \cdot \sqrt{2gh}\). Here, \(h\) is the height of water, \(V = h \cdot 1^2 = h\), and area of the tank bottom is \(1 \text{ m}^2\). Therefore, \(\frac{dh}{dt} = -A \cdot \sqrt{2gh} = -0.0016\pi \cdot \sqrt{2gh}\).

Step 4: Substitute Known Quantities into the Equation

Substitute the proportionality constant into the equation: \(\frac{dh}{dt} = -0.0016\pi \cdot \frac{5}{\pi} \cdot \sqrt{2gh} = -0.008 \cdot \sqrt{2gh}\).

Step 5: Solve the Differential Equation

Rearrange to separate the variables: \(\frac{dh}{\sqrt{h}} = -0.008 \cdot \sqrt{2g} \cdot dt\). Integrate both sides. The integral on the left is \(2\sqrt{h}\). Thus, \(2\sqrt{h} = -0.008 \cdot \sqrt{2g} \cdot t + C\).

Step 6: Apply Initial Conditions

At \(t = 0\), \(h = 1\), giving \(2 \cdot 1 = C\). Therefore, \(C = 2\). Use this to rewrite the equation: \(2\sqrt{h} = -0.008 \cdot \sqrt{2g} \cdot t + 2\).

Step 7: Solve for the Time When the Tank is Empty

Set \(h = 0\) to find when the tank is empty: \(2 \cdot 0 = -0.008 \cdot \sqrt{2g} \cdot t + 2\). Solve for \(t\). Rearranging gives \(-2 = -0.008 \cdot \sqrt{2g} \cdot t\) thus, \(t = \frac{-2}{-0.008 \cdot \sqrt{2g}}\).

Step 8: Calculate the Specific Time

Substitute \(g = 9.8 \, \text{m/s}^2\), \(t = \frac{2}{0.008 \cdot \sqrt{2 \cdot 9.8}}\). Calculate the value, resulting in \(t \approx 226.77 \, \text{seconds}\).

Key Concepts

Differential Equation in Torricelli's Law

When dealing with fluid dynamics, differential equations are powerful tools to model the changes over time. Specifically, in Torricelli's Law, we use a differential equation to represent how the height of the water in a tank decreases as water flows out. This kind of problem involves variables that change continuously, such as water height and time. In the case of Torricelli's Law, the differential equation is set up based on the principles of fluid dynamics. This equation often comes in the form:\[\frac{dh}{dt} = - A \cdot \sqrt{2gh}\]Here, \( \frac{dh}{dt} \) represents the rate of change of the water height, \( A \) is the cross-sectional area of the hole, and \( g \) is the acceleration due to gravity. The negative sign indicates that the height is decreasing over time. By solving this equation, one can determine how long it takes for all the water to drain from the tank.

Understanding Fluid Dynamics

Fluid dynamics is the study of the flow of liquids and gases. Torricelli's Law is a principle within this field that describes the speed at which a liquid exits an orifice. In our problem, fluid dynamics principles help us determine how the water flows out of the tank through a hole.**Important Concepts in Fluid Dynamics:**- **Flow Rate:** The volume of fluid that passes a point per unit time. In our case, it's governed by the size of the hole and the speed of outgoing water.- **Velocity at the Orifice:** Given by Torricelli's Law as \( v = \sqrt{2gh} \), meaning the exit speed of water is dependent on the height of water above the hole.- **Impact of Gravity:** Gravity plays a crucial role in fluid dynamics by influencing the speed at which the liquid exits. This gravitational force is reflected in the equation as \( g \), a constant in motion equations.

Role of Integration in Calculus Problem-Solving

Integration is a key mathematical tool used to solve differential equations like the one in Torricelli's Law. It helps to find the function that describes the systems behavior over time. To determine the time it takes for the tank to completely drain, integration allows us to piece together the continuous change in water height and solve for time.In our exercise, the equation is:\[2\sqrt{h} = -0.008 \cdot \sqrt{2g} \cdot t + C\]By integrating both sides of the differential equation, we arrive at a relationship between the water height \( h \) and time \( t \), capturing how the system evolves. Solving this integral is crucial because it transforms the rate of change equation \( \frac{dh}{dt} \) into a usable function of time.

Step-by-Step Calculus Problem Solving

Solving complex problems like the one involving Torricelli's Law demands a clear, step-by-step approach. Here, calculus is not just about solving the equations; it’s about understanding the physical scenario and translating it into the language of mathematics.**Steps to Solve the Draining Problem:**

• Start by setting up the differential equation that describes the change, derived from physical laws.
• Use known constant values, like the proportionality constant and gravitational acceleration.
• Integrate the equation to find a function relating time and water height.
• Apply initial conditions (e.g., \( h = 1 \) at \( t = 0 \)) to solve for constants of integration.
• Rearrange the resulting equation to solve for the desired variable, in this case, the time when \( h = 0 \).

By following these clear steps, we arrive at the crucial result: how long it takes for the tank to drain based on the initial setup and physical laws.