# Rate of flow - height of water problem

1. Dec 4, 2007

### ineedmunchies

Ok so I've been given this problem for one of my classes (see attatchment) :

I haven't done much with differential equations before and I'm a bit stuck. What has been putting me off is the fact that it says the vessel has a constant cross sectional area A. which I am guessing is the grey shaded area.

I don't understand how whether the differential equation is meant to include this A term. Surely it can't be constant if the height of the water varies?

Any help would be greatly appreaciated.

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2. Dec 4, 2007

### dst

You have two of the dimensions in volume accounted for, by the constant cross sectional area. The grey shaded area is the liquid. The cross sectional area represents the 2 dimensions of the cylinder aside from the height which is changing. Don't be fooled.

You shouldn't have to include the cross sectional area because the question asks for a function of height. Now, if the cross-sectional area was not constant, you would have to include a derivative of that as well.

Last edited: Dec 4, 2007
3. Dec 4, 2007

### ineedmunchies

ah thank you, well in that case would it be something like:

$$\frac{dq}{dt}$$ = A $$\frac{dh}{dt}$$

4. Dec 4, 2007

### dst

No, you shouldn't include A, because it's a constant. The question concerns only changing quantities that affect that specific variable h. h isn't going to be changed with respect to A because there is no change in A, i.e. dA/dT = 0.

In this case, there is obviously only one change, a linear increase in h(t) coming from q(t).

5. Dec 7, 2007

### coomast

ineedmunchies, you were almost correct in the equation, but made an error which you could avoid if you had checked the units. You have for the flow entering the vessel q expressed in m^3/s and the cross-sectional area A in m^2, therefore you have for the change in height:

$$\frac{dh(t)}{dt}=\frac{q(t)}{A}$$

This is the differential equation you need to solve. Remember you need the flow q(t) as a function. It can be a constant, but that was not given in the original post, as wasn't the height h(t) at a certain time t. These things will make it completely solvable. It is a basic equation, but in case you have any problems, post them.