Modeling with differential equations

In summary: C:4 = [(2αa*sqrt(2g*4))/(πr^2)]*0 + CC = 4Now, we can plug in all the values we know (r = 2 ft, A = 1 in^2, α = 0.6, g = 32.2 ft/s^2) and solve for h when t = 300 seconds:h = [(2*0.6*1*sqrt(2*32.2*4))/(π*2^2)]*300 + 4h ≈ 0.43 feetIn summary, using the formula for the rate of outflow and the volume of
  • #1
tracedinair
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Homework Statement



A hemispherical bowl 4 feet in diameter is full of a liquid. Show that the bowl empties through an orifice in the bottom, of one square inch effective cross section, in a little than five minutes.

Homework Equations



The Attempt at a Solution



My problem is where to even really begin. My guess is to start with the formula equation rate of outflow and rate of change of liquid, A(h)(dh/dt) = -αa*sqrt(2gh). But where do I go after that?

Edit: I just figured it out. It's just a variable separable. Sorry. n/m
 
Last edited:
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  • #2
Hi there,

Thank you for reaching out for help with this problem! Let's break it down and see if we can come up with a solution together.

First, let's define some variables and constants:
- r = radius of the hemispherical bowl (in feet)
- h = height of the liquid in the bowl (in feet)
- A = cross-sectional area of the orifice (in square inches)
- α = coefficient of discharge (a constant that takes into account the shape and roughness of the orifice)
- g = acceleration due to gravity (32.2 ft/s^2)

Now, let's write down the equations we know:
- Volume of liquid in the bowl = Volume of liquid that has flowed out through the orifice
- Volume of liquid in the bowl = (1/2)πr^2h
- Volume of liquid that has flowed out through the orifice = A(h)(dh/dt)

Since the volume of liquid in the bowl is decreasing, the rate of change of the liquid's volume with respect to time is negative. Therefore, we can write:
d[(1/2)πr^2h]/dt = -A(h)(dh/dt)

Next, we can use the equation for the rate of outflow given in the problem statement:
αa*sqrt(2gh) = A(h)(dh/dt)

Now, we have two equations with two unknowns (h and dh/dt). We can solve this system of equations by substituting the second equation into the first one:
αa*sqrt(2gh) = (1/2)πr^2h * dh/dt

We can rearrange this equation to get dh/dt by itself:
dh/dt = (2αa*sqrt(2gh))/(πr^2)

Now, we can integrate both sides with respect to t from t = 0 to t = 5 minutes (or 300 seconds):
∫ dh/dt dt = ∫ (2αa*sqrt(2gh))/(πr^2) dt
∫ dh = ∫ (2αa*sqrt(2gh))/(πr^2) dt

Integrating both sides gives us:
h = [(2αa*sqrt(2gh))/(πr^2)]*t + C

Since we know that h = 4 feet when t = 0, we
 

What is the purpose of using differential equations in modeling?

The purpose of using differential equations in modeling is to describe and predict the behavior of a system over time. Differential equations can capture the relationship between different variables and how they change in relation to one another, allowing for a more accurate representation of complex systems.

What are some common applications of modeling with differential equations?

Modeling with differential equations is used in many fields, such as physics, biology, economics, and engineering. Some common applications include predicting population growth, understanding the spread of diseases, analyzing chemical reactions, and designing control systems for machines.

What are the steps involved in modeling with differential equations?

The first step in modeling with differential equations is defining the variables and their relationships, usually through a set of equations. Then, the equations are solved using mathematical techniques to determine the behavior of the system. The results are then compared to real-world data to validate the model and make adjustments if necessary.

What are the limitations of modeling with differential equations?

One of the limitations of modeling with differential equations is that they may not always accurately represent real-world systems. This is because some systems may have complex or unknown factors that cannot be captured in a simple set of equations. Additionally, the accuracy of the model depends on the accuracy of the initial conditions and parameters used.

Are there alternative methods for modeling besides using differential equations?

Yes, there are alternative methods for modeling, such as using computer simulations or machine learning algorithms. These methods may be more suitable for complex systems that cannot be accurately represented by a set of equations. However, differential equations are still widely used and are often preferred due to their simplicity and ability to provide analytical solutions.

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