Rate of Change and Water Coolers

There's a water cooler in my camp's building and this week while I was stuck in camp I wondered about how I could calculate the flow rate of the water cooler, similar to the discharge of a river.

While filling up my water bottle, the first thought that entered my mind was that the height of the water in the bottle is increasing.

RATE OF CHANGE.

So I began writing things down.

Let the height of the bottle be h, and let the diameter of the bottle be d. (Sorry, tau advocates, I love tau too but the engineers are right, the diameter is easier to measure.)

So the volume:

We're aiming to find the change of volume with respect to time.

So we want to find this:

So now we need two things, how the height can affect the volume, and how the time taken can affect the height.

After getting this general form, I then proceeded to measure my bottle, and the time taken to fill it completely.

At this point I realised I had a very easy way to check my results. My bottle is graduated, and has a capacity of one litre. Coupled with the time taken to fill it, this already shows that the flow is about 50cc per second.

The volume calculation isn't exactly 1000cc because my bottle is slightly concave, which the height of 17cm didn't account for. Even so, it is still pretty accurate, with a percentage error of 8%.