Fermi Contest Solution…

Thank you to everyone who entered our Fermi Estimation Contest this past January. Congratulations to winner Alex D, who received a free copy of How Many Licks? Several contestants asked for my solution, so I’m posting my results below:

Problem:

How many snowflakes are in this snowman?

Solution:

I assumed the snowman was made of three spheres of radii R₁=30 cm, R₂=45 cm, and R₃=65 cm. The volume of the snowman is then given by:

V = (4Pi / 3)*(R₁³ + R₂³+R₃³) ~ 1.6m³

From Hypertextbook.com, there are roughly 100 crystals per snowflake and 10¹⁸ molecules per crystal. One water molecule weighs 2.992x10^-26 kg. From these facts, we can calculate the mass of a flake:

(100 crystals per flake)

*(10¹⁸ molecules per crystal)

*(2.992x10^-26 kg per molecule)

-----------------------------------

3.0x10⁶ kg per snowflake

According to Wikipedia, the density of snow varies considerably from 8% the weight of water for newly fallen snow, to 30% after it settles, to 50% in late spring. Since the density of water is about 1000kg/m³ and the snowman is pretty densely packed, I chose 500 kg/m³.

Using these numbers, we can calculate the number of snowflakes:

(1.6 m³)*(500 kg per m³) / (3.0x10⁶ kg per snowflake) ~ 2.7x10⁸ snowflakes