Leaky Bucket and total volume

Question

I’m working on a problem set and i’m just stuck on a question.

You have a leaking bucket. When the bucket is full, it loses 1L of water in a minute. After 7 minutes, the bucket is half empty. What is the volume of the bucket knowing that the speed at which it drains depends on how full it is ?

I’ve tried to model it with an ODE but to no avail, thank you if you can help !

Is the rate of water loss supposed to change depending on how full it is? Because otherwise, this doesn’t require an ODE. — Thomas Andrews, May 27 '21 at 22:22
Yeah I worked until there, but then, $V(t) = cste^{-kt}$, Hence $V(0) = cst , V(7) = \frac{cste^{-7k}}{2}$, and it seems like something is missing for me to find the constant ? — user932635, May 27 '21 at 22:57
Well, I wouldn’t be duplicating your work if you had shown your work. — Thomas Andrews, May 27 '21 at 23:03
My mistake I should've shared this from the get go - but I was unaware of whether or not it was the right direction, thank you — user932635, May 27 '21 at 23:06
Your $V(7)$ formula is wrong. You want $V(7)=\frac{V(0)}{2},$ but the differential equation says it is... — Thomas Andrews, May 27 '21 at 23:16

Thomas Andrews · Accepted Answer · 2021-05-27T23:08:47.493

1

The most obvious equation for the rate of the leak, if it depends on $V,$ the volume of water, is:

$$\frac{dV}{dt}=-\frac{V}{V_0}$$ where $V_0=V(0)$ is the full volume for the bucket, and we wanted $\frac{dV}{dt}(0)=-1\, \text{liters/min}.$

You get $V(t)=V_0e^{-t/V_0}$ and you want to solve:

$$V(7)=\frac12 V_0.$$

edited May 27 '21 at 23:08

answered May 27 '21 at 22:57

Thomas Andrews

177,126

That's it thank you very much ! – user932635 May 27 '21 at 23:21

Leaky Bucket and total volume

1 Answers1