Math Is Fun Forum

Reuel

Member

Registered: 2010-11-28

Posts: 178

The Flow of Water

"Assuming the flow of water (its volume per unit of time) through a hole in the bottom of a tank is proportional to the product of the area of the hole and the square root of the depth of the water. Let

where h(ft) = depth of water, A(ftxft) is the area of the water's surface at time t in seconds, and B(ftxft) is the area of the hole. The constant of proportionality k depends on the condition.

a. Find the time required to empty a cubical tank whose edge is 4 feet long. The tank has a hole 2 inches in diameter in the bottom and is initially full. In this case, let k = 4.8.

b. A funnel with the shape of a right circular cone and with vertex down and is full of water. If half the volume of water runs out in time t, how long to completely empty?"

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: The Flow of Water

hi Reuel,

Assuming the flow of water (its volume per unit of time) through a hole in the bottom of a tank is proportional to the product of the area of the hole and the square root of the depth of the water

I would have put

For a cube, let

then (as A is constant)

t = 0, h = 4

When is h = 0 ?

For the cone let R be the radius at the top, r the radius at any other time

and let H be the height of the cone, and h the height at any other time

but

so

I think life will be simpler if I put all these constants together as one single constant, D

let

then

h = H when t = 0

Empty when h = 0, when

So all I've got to do is replace D, get the time (t') at half volume, cancel all the unknowns (I hope ) and get the final time in terms of t'.

Easy ?  I need to put my head in a bucket of ice for a while first !   at 21.01 GMT

Oh   I think I'll need a bigger bucket!  21.19 GMT

Arrhhh..... That's better.

using (i)

As you can see, I had some difficulty getting my head around the end of this problem.  IT MAY STILL HAVE A MISTAKE!  I need a break from it.  I'll review what I've done tomorrow, when I'm fresh to it.

LATER EDIT.

It is now Thursday, at 13.21 GMT.

I've looked through the whole document and cannot find any errors.. That doesn't mean there aren't any! 

Bob

Last edited by Bob (2011-03-03 01:22:34)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: The Flow of Water

Whoa... thanks for the full reply. Let me read what all you said.

First of all, the initial differential equation was given in the problem. Does it help to change it to the form you put it in?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,627

Re: The Flow of Water

hi Reuel,

I thought that DE had been given but I think it only works for part a.

For part b, the area is constantly changing which is why I developed my own DE based on the first sentence.

In part a , A is constant, which leads to the given DE.

I don't think it can work for part b.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Reuel

Member

Registered: 2010-11-28

Posts: 178

Re: The Flow of Water

Thanks bob. Sorry to not respond right away, I've been distracted. I appreciate your help. Setting these things up is the hard part, for me.

I have a few other word problems but I am going to work on them myself, first.