# Torrcellis law differential equations

• mikky05v
In summary, the problem statement is that the person is stuck and needs help to solve differential equations. The given/known information is that they need to use the formula r = 3h/5 in part (c) and that the "actual" time is longer if they solve it using r = .59h + .5. The problem statement also states that part (e) needs the formula r = -3/5 h +30 which gives the radius at any given height. The given/known information is that they need to find the constant c by taking t=0 and h=50.
mikky05v
1. The problem statement, all variables and given/known
This is a group project for differential equations. I ended up without a group, lucky me. I've been trying to work through this on my own but I am stuck. Sorry about the pictures, typing it all out would of taken ages.

http://imgur.com/qeOkl3n
http://m.imgur.com/uLZWRCX
http://imgur.com/z3SC4st

## Homework Equations

He sent us the email
Project C:  you can get an approximate answer for part (d) by using the formula r = 3h/5 in part (c)  [so A(h) = Pi*(3h/5)^2]  you really should be able to solve that diff eq in (c) with that info.  (note that the "actual" time is a bit longer if you solve it the "correct" way using r = .59h + .5).  For part (e)--you first need the formula r = -3/5 h +30 which of course gives you the radius at any given height.  This makes your A(h) = Pi* (-3/5 h + 30)^2.  When you put this into the differential equation and divide by the square root of h and then integrate you should get an equation with h^5/2 and h^3/2 and h^1/2 and t (and of course coefficients on all of those!).  Once you find your constant by inputting your initial condition [h(0)=50] you can solve the resulting implicit equation BUT you can't do it by hand, you have to use technology!  (I used desmos by inputting "x" for my "t" and "y" for my "h")  the graph then showed me the time to drain the other tank is around 10 minutes (600 seconds).  I'll leave it to you to find the exact value.

## The Attempt at a Solution

l did part a by integrating twice.

b. I am not entirely sure what this section wanted. I put A (h) dh/dt= -a(sqrt(2gh))

C. r=3h/5 according to email
A (h) = pi (3h/5)^2
a= pi (1/2) ^2 = pi/4
g= 98.1 cm/s^2
Giving the seperable differential equation
Pi (3h/5)^2 dh/dt = -pi/4 sqrt (2×98.1h)
That simplifies to .0257012h^(3/2) dh=dt

D. Integrating both sides I got t= .0102805h^(5/2) + c
I thought to solve for c by taking t=0 and h=50 but I get c=-181.7352816 so I know I did something wrong.
Thats as far as I've gotten.

The top of the second page got cut off. Could you make a different copy of this page and post it?

http://m.imgur.com/uLZWRCX

Try turning off the toolbar at the top of the page before making the image.

The 2nd page is just the bottom of the first page, to see the stuff above it look at the first page. :)

has anyone had a chance to look at this yet, i keep doing circles around the same thing. I know my equation is wrong but I can't figure out what to do differently

## 1. What is Torricelli's law differential equation?

Torricelli's law differential equation is a mathematical equation that describes the rate of change of fluid flow through an orifice in a container. It is named after Italian physicist Evangelista Torricelli, who discovered the law in the 17th century.

## 2. How is Torricelli's law differential equation derived?

Torricelli's law differential equation is derived from Bernoulli's equation, which relates the pressure, velocity, and height of a fluid in a container. By applying conservation of mass and simplifying the equation, we can obtain the differential equation for fluid flow through an orifice.

## 3. What is the significance of Torricelli's law differential equation?

Torricelli's law differential equation is significant in fluid dynamics as it allows us to predict the rate of fluid flow through an orifice. This is useful in various engineering applications, such as designing pipes, nozzles, and pumps.

## 4. What are the assumptions made in Torricelli's law differential equation?

The main assumptions made in Torricelli's law differential equation are that the fluid is incompressible, the flow is steady, and there is no external force acting on the fluid. It also assumes that the fluid is ideal, meaning it has no viscosity or turbulence.

## 5. How is Torricelli's law differential equation used in real-life situations?

Torricelli's law differential equation is used in various real-life situations, such as designing water supply systems, irrigation systems, and hydraulic engineering projects. It is also used in weather forecasting and studying the behavior of gases in containers.

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