Modeling with differential equations

[tracedinair]

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

 

[mighty2000]

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