## Raindrops

### June 27, 2014

We’re going to try something different today. We often have interview questions here, but always of the type that require writing a program. Today, we will have one of the brain-teaser type of interview question:

How many raindrops fall on the planet every year?

Your task is to estimate the answer to the question posed above. When you are finished, you are welcome to read a suggested solution or to discuss your solution in the comments below.

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Let’s say that the earth is approximately a 10000km cube and that a cube has approximately 10 sides, that gives an area of 10^15 square metres, so with 100cm of rain per year, that’s 10^14 cubic metres of water. A raindrop is about a 5mm cube, so that’s 200^3 in a cubic metre, which is round about 10^7, so total drop count is 10^21. Or thereabouts…

Average raindrop size seems to be the big unknown here.

Apparently, average rainfall is more like a metre a year, and smaller raindrops are presumably the majority, so maybe 10^23 or even higher is probably closer.

back of the envelope calculation based on vaguely remembered factoids and only carrying one significant digit (using ~ to mean approximately)

earth radius ~ 4000 mi * 5000 ft/mi * 12 in/ft * 3 cm/in ~ 720e6 ~ 7e8 cm

earth surface area = 4 * pi * r^2 ~ 4 * 3 * 50e16 = 60e17 cm^2 = 6e18 cm^2

assume everywhere is same as rainiest place I know of (one of the Hawaiin islands), then scale at the end:

rainiest place ~ 500 in rain/yr * 3 cm / in = 1500 ~ 2e3 cm rain / year

total volume of rain = 2e3 cm / year * 6e18 cm^2 = 12e21 ~ 1e22 cm^3 / year

rain drop ~ 2.5 mm radius ( they split if much more than 5 mm) ~ .3 cm

volume of raindrop = (4/3) pi r^3 ~ (4/3)*3*(.3^3) ~ 4 * .027 ~ .1 = 1e-1 cm^3

number of raindrops = total volume of all rain / volume of one drop

~ (1e22 cm^3 / year) / (1e-1 cm^3) = 1e23 rain drops per year. Hmm, almost Avogadro’s number, a mole of raindrops per year ;-)

if the rainiest place gets 100 times as much rain as the world average, there would be around

1e21, or 1,000,000,000,000,000,000,000, raindrops per year.Here’s a similar discussion about how many snowflakes it would take to get 6 feet of snow on earth: http://what-if.xkcd.com/104/

About the same as everyone else

Estimated earth circumference: 24,000mi (actual 24,901) error 3.6%

Estimated annual rainfall: 1meter (actual .99m precipitation) error 1%

Estimated volume of a raindrop: 0.2mL (No real data on this but from a few random places this seems in the ball park)

I forgot what the surface area of the sphere was but I remembered the volume was 4/3pi*r^3 and I knew the derivative of volume is surface area so I got to 4pi*r^2

After fixing a few zero issues, didn’t use a calculator, i ended with 1.9*10^21. Using the actual numbers and the 5 drops per mL you get 2.5*10^21 or an error of 25%.

Now the precipitation also includes snow and the problem just said raindrops so should probably take out 30%? from that.

Pretty incredible that a group of people can estimate such a figure that initially seems impossibly large and all get so close.