Any kid who's ever been entranced by an advent wreath knows that a tapered advent candle shrinks faster on Sunday night when it's new and slender than on Saturday night when it's old, stubby and caked with melted wax.  

How much faster?  As an apropos application of math during this Christmas season, let's derive a formula for the height of a burning candle as a function of time.

Assuming the candle has the shape of a cone when it is new and that it drips wax at a constant rate as it burns, we show that the height of the candle shrinks roughly in proportion to the cube root of time.

We model our partially burned candle as a truncated cone (or frustrum for all you Lateinophiles)

Note that , and   

The volume of the frustrum is the volume of the new candle minus the volume that has already burned.  This is given by

We can eliminate  using the following proportion, which holds by similar triangles.

Substituting (3.3) into (3.1), we get

But since we assumed the candle loses volume at a constant rate k, the second term above is equal to .  

Thus, we have the following volume equation, which we can then solve for

Thus, the height of the candle shrinks in proportion to the cube root of time.

Let's also examine the rate of shrinking.

 =

This expression makes intuitive sense.  First of all, it's negative, meaning the height never increases.  More importantly, it predicts that the taller and thinner the candle , the faster its height shrinks with time.  This is what every kid knows.

Notice that for a fixed height and burn rate, decreasing the diameter makes the graph steeper, meaning the candle's height will shrink more quickly.