Math CentralQuandaries & Queries


Question from Stella, a student:

Liquid is being pored into the top of a funnel at a steady rate of 200cm^3/s. The funnel is in the shape of an inverted right circular cone with a radius equal to its height. It has a small hole in the bottom where the liquid is flowing out at a rate of 20cm^3/s. How fast is the height of the liquid changing when the liquid in the funnel is 15cm deep?
At the instance when the height of the liquid is 25cm, the funnel becomes clogged at the bottom and no more liquid flows out. How fast does the height of the liquid change just after this occurs?

Hi Stella.

The second part of the question looks easier than the first but it isn't.

In the second part of the question, you only have liquid flowing one way (into the funnel), but you can see in the first part of the question there is a flow into the funnel and another one out. However, they are both constant flow rates and don't depend on the height, so you can lump them together. Thus the first part of the question is equivalent to an constant inbound flow rate of 180 cm^3/s.

The flow rate is volume per unit time. The rate you are interested in is distance (height) per unit time. So you need something that connects the height of the liquid to the volume.

A right circular cone, inverted in this manner with height h equal to the radius has the volume v given by:

v = π h3 / 3.

So when we differentiate with respect to time, we get an equation involving dv/dt (that's the flow rate) and dh/dt (that's what you want: the vertical speed of the water level).

d/dt (v) = d/dt (π h3 / 3)

Constant factors come out:

dv/dt = (π / 3) d/dt (h3)

And the exponent rule says to bring down the 3 from h3 as a factor and reduce the exponent by 1, so we get 3h2, leaving just the dh/dt:

dv/dt = (π / 3) (3h2) (dh/dt)

So we've done the calculus part. Now we just solve for what we want (dh/dt):

dh/dt = (dv/dt) / (h2π).

So given a value for dv/dt (the net flow rate) and a particular height h that we are interested in, we can find the change in height over time (that is the vertical speed of the water level).

Stephen La Rocque.

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