Rate of flow - height of water problem

[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.

 

[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.

 

[ineedmunchies]

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

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

 

[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).

 

[coomast]

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

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

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.