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