Help needed: interesting calculus problem

[jimi]

'Fluid enters a leaky vessel through a valve. The valve admits fluid at a rate proportional to the volume of fluid already in the vessel, and the rate of leakage is proportional to the square of the volume already in the vessel. There is a balance between inflow and outflow when the volume in the vessel is Vo. Initially there is a volume Vo/4 in the vessel, and the volume increases to Vo/2 in time T. Find the time taken for the volume to increase from Vo/4 to 3Vo/4.'

I worked this out two ways and got exactly the same answer, but as one of the methods is only supposed to be valid for small changes I'm puzzled because there does not appear to be any assumptions about the size of the changes in my calculations. Maybe I'm missing something or went wrong somewhere. I can try to post my calculations if you want me to. Thanks for any help.

 

[stapel]

Maybe I'm missing something or went wrong somewhere. I can try to post my calculations if you want me to.

If you're wanting somebody to evaluate the validity of what you've done, then, yes, it might be helpful if you showed what that was. Thank you.

Eliz.

 

[jimi]

$V/$t = dV/dt in the limit as t approaches 0 ($ means 'a small change in')
so:
$V/$t ~ dV/dt when t is small (~ means approximately equals)
and:
$V ~ $t(dV/dt)

dV/dt = aV - bV^2 (where a and b are constants)
when volume = Vo, inflow = outflow, i.e. aVo = bVo^2
Vo = a/b

The volume increases from Vo/4 to Vo/2 in time T so, using
$V ~ $t(dV/dt):
Vo/4 ~ T(aVo/4 - b(Vo/4)^2)
substituting a/b for Vo:
a/4b ~ T(a^2/4b - a^2/16b)
a/4b ~ T(3a^2/16b)
T ~ 4/3a

The time t taken for the volume to increase from Vo/4 to 3Vo/4:
Vo/2 ~ t(aVo/4 - b(Vo/4)^2)
a/2b ~ t(3a^2/16b)
t ~ 8/3a

The time taken is therefore, approx., 2T.
The other way I solved the differential equation and got the same answer. I thought this would give me a different answer because it does not make any assumptions about the change in V or t being small.

 

[jimi]

Hello stapel? I thought you were going to help me.

 

[stapel]

Hello stapel? I thought you were going to help me.

I only suggested that, if you wanted your work to be checked, it would be helpful if you posted it. My reply wasn't meant to be a guarantee that I would do the checking. I apologize for the confusion.

Eliz.