Problem 34 - Advanced Engineering Mathematics

Question

A tank in the form of a right-circular cylinder of radius 2 feet and height 10 feet is standing on end. If the tank is initially full of water, and water leaks from a circular hole of radius \(\frac{1}{2}\) inch at its bottom, determine a differential equation for the height \(h\) of the water at time \(t\). Ignore friction and contraction of water at the hole.

Step-by-Step Solution

Answer

The differential equation is \( \frac{dh}{dt} = -\frac{1}{2304} \sqrt{2gh} \).

Step 1: Understanding the problem

We have a cylindrical tank with a radius of 2 feet and height 10 feet. Initially, this tank is full of water. Water is leaking out of the tank through a hole with a radius of \(\frac{1}{2}\) inch located at the bottom. We need to set up a differential equation to describe the height \(h\) of the water at any time \(t\).

Step 2: Converting units for the hole radius

The radius of the hole is given in inches, so we convert it to feet for consistency in units. Since there are 12 inches in a foot: \(\frac{1}{2}\) inch = \(\frac{1}{2 \times 12}\) feet = \(\frac{1}{24}\) feet.

Step 3: Apply Torricelli's Law

According to Torricelli's Law, the speed \(v\) of efflux of a fluid under gravity through a hole is given by \(v = \sqrt{2gh}\), where \(g\) is the acceleration due to gravity and \(h\) is the height of water. For water, \(g = 32.2 \text{ ft/s}^2\).

Step 4: Relate velocity to flow rate

The flow rate \(Q\) of water through the hole can be modeled by the equation \(Q = A_h v\), where \(A_h\) is the area of the hole. Given the radius of the hole is \(\frac{1}{24}\) feet, we have: \(A_h = \pi \left( \frac{1}{24} \right)^2\). Then \(Q = \pi \left( \frac{1}{24} \right)^2 \sqrt{2gh}\).

Step 5: Volume of water in the tank

The volume \(V\) of water in the tank at height \(h\) is given by the formula for the volume of a cylinder: \(V = \pi r^2 h\), where \(r\) is 2 feet. So, \(V = 4\pi h\).

Step 6: Relate flow rate to volume change

The rate of change of volume \(\frac{dV}{dt}\) is equal to \(-Q\) (negative sign indicating water is exiting). So, we have: \(\frac{d(4\pi h)}{dt} = -\pi \left( \frac{1}{24} \right)^2 \sqrt{2gh}\).

Step 7: Formulating the differential equation

Simplifying \(\frac{d(4\pi h)}{dt} = -\pi \left( \frac{1}{24} \right)^2 \sqrt{2gh}\) gives \( 4\pi \frac{dh}{dt} = -\pi \left( \frac{1}{576} \right) \sqrt{2gh}\). Thus, \( \frac{dh}{dt} = -\frac{1}{2304} \sqrt{2gh}\). This is your differential equation for the height of the water.

Key Concepts

Torricelli's Law

Torricelli's Law is a fundamental principle that describes the speed at which fluid exits a hole under the influence of gravity. This law was discovered by Evangelista Torricelli, a 17th-century Italian physicist and mathematician. According to this principle, when a fluid flows out of a hole in a container, its speed is determined by the height of the fluid above the hole.
Mathematically, this can be expressed as \( v = \sqrt{2gh} \), where:

• \( v \) is the velocity of the fluid exiting the hole.
• \( g \) is the acceleration due to gravity, typically \( 32.2 \text{ ft/s}^2 \) for these calculations.
• \( h \) is the height of the fluid above the hole.

Torricelli's Law assumes ideal conditions without friction or air resistance, simplifying computations in applications like fluid dynamics problems. When applied to our problem, it determines how fast water leaks from the cylindrical tank based on the water level height at any given time.

cylindrical tank

A cylindrical tank is a simple and common storage structure in both industrial and domestic settings, often used for holding liquids like water or fuel. It has a circular base and a specific height, forming a three-dimensional shape resembling a soup can.
The mathematical description of a cylindrical tank involves defining:

• The radius \( r \) of the circular base.
• The height \( h \) of the cylinder.

For the exercise, our tank features a radius of \( 2 \) feet and a height of \( 10 \) feet. With these dimensions, you can easily calculate properties like the volume using the formula for the volume of a cylinder: \( V = \pi r^2 h \). In this case, the volume is \( 4\pi h \) cubic feet, where \( h \) is the current height of the water in the tank.
Understanding the dimensions of the tank is crucial for setting up the differential equation we're working with.

water flow rate

The water flow rate refers to the volume of water passing through a point, such as the hole at the bottom of the tank, per unit of time. It is critical for quantifying how quickly the water level is changing in the tank.
The flow rate \( Q \) can be calculated using the formula \( Q = A_h v \), where:

• \( A_h \) is the area of the hole through which the water exits. For a hole radius of \( \frac{1}{24} \) feet, \( A_h = \pi \left( \frac{1}{24} \right)^2 \).
• \( v \) is the velocity of the water exiting the hole, given by Torricelli's Law as \( \sqrt{2gh} \).

By knowing the flow rate, we can relate this to how fast the water level in the cylindrical tank drops over time, thus enabling us to express the dynamics through a differential equation.

volume change rate

The volume change rate is an essential aspect of understanding how the water volume within the tank decreases over time due to the outflow through the hole. This rate of change forms the basis for setting up a differential equation in the problem.
Since the volume of water in the tank is defined by \( V = 4\pi h \), the rate of change of this volume is expressed as \( \frac{dV}{dt} \). Due to water leaking out, this rate is negative and relates to the flow rate as \( \frac{dV}{dt} = -Q \).
Simplifying this in the exercise, we derive the differential equation relating to the height \( h \) of the water:

• The rate of volume change \( \frac{d(4\pi h)}{dt} = -\pi \left( \frac{1}{24} \right)^2 \sqrt{2gh} \).
• By simplifying, the equation becomes \( 4\pi \frac{dh}{dt} = -\pi \left( \frac{1}{576} \right) \sqrt{2gh} \).
• Therefore, \( \frac{dh}{dt} = -\frac{1}{2304} \sqrt{2gh} \), which succinctly represents the relationship between the rate of height change and the dynamics of water flow.