- #1

Mike T

- 2

- 0

## Homework Statement

I've been asked to solve a problem where a bathtub is being filled at a constant rate, meanwhile water is draining out continuously. The bathtub fills with a constant flow rate of [itex] \frac{dI}{dt} [/itex] and drains with a flow rate [itex] \frac{dO}{dt} [/itex] which is directly proportional to √h where h is the height of the water.

The question is to come up with an equation describing the height of the water over time, given this constant input and varied output. Use this to find out what time the bath water will overflow.

## Homework Equations

[itex] netflow = \frac{dI}{dt} - \frac{dO}{dt} [/itex]

3. The Attempt at a Solution

3. The Attempt at a Solution

So I've assumed that I should come up with a an equation in the form [itex]h(t)[/itex] where h is the height, and t is the time variable.

I know the volume of the bath, so what height needs to be reached. I've tried integrating both of the flow rates, and then combining them e.g. [itex]h = I(x) - O(x)[/itex], and then solving for the initial condition h = bath height. This gives me the time to fill the bath (although I don't now how accurate it is!) - how I can I make this into a more general solution to find the height at any time t?

I'm having great difficulty relating the input and output in this equation together, and would welcome any points on the general approach, as well as my current approach above.