A Reasonable Approximation

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How Red Will the Oceans Run?

In which I inexpertly channel Randall Munroe

Soon the seas will turn red with the blood of the human race, as the unspeakable terrors come from beyond the gate, which is Yog Sothoth, to devour all in their path! Ia! Shub Niggurath! Ia! Ia!

Who will be eaten first?

If you were to mix the blood of the human race into the oceans, how red would they turn?

A quick fermi estimate: there are seven billion humans, with about eight pints of blood each, which is about five litres. So we have about 35 billion litres of blood to work with.

How about the oceans? They cover two thirds of the earth's surface. I know that the surface of the earth is $4πr^2$ where $r$ is 6,400 km. $(64 × 10^5)^2 = 2^{12} × 10^{10} = 4096 × 10^{10}$. $4π$ is approximately ten, so the surface area of the earth is about $4×10^{14}$ m². Two thirds of that is $2.5 × 10^{14}$. I don't know how deep they are, but 100m seems like a reasonable guess at an average. That gives us about $2.5 × 10^{16}$ m³ of ocean.

A cubic metre is a thousand litres, so we have about 25 billion billion litres of ocean. For every drop of blood, we have almost ten billion drops of ocean.

I don't know how blood and water mix, but I'm confident that the oceans will not be running red with blood any time soon, on average. Sorry cultists.

Fact checking: Wolfram Alpha tells me that there are actually about 10¹⁸ m³ of ocean, which is 40 times what I guessed, which is okay for a Fermi estimate. The oceans are still not running red with blood.

NOAA, whoever they are, tell me that the average depth of the ocean is 4km, which is also 40 times what I guessed. That's not great for a Fermi input.

(Initially, I forgot the $4π$ factor in the surface area of the earth, putting me off by 400 times. I'm not sure how to judge myself on Fermi estimates when I get somewhat-reasonable inputs and do the math wrong.)

However! The quote specifically mentioned seas. I've cheated by counting oceans as seas. What if we exclude them? I'm going to do this just by listing the oceans and subtracting their volume from the total. We have:

It's possible that this double-counts the Southern ocean, since some people divide it up between the Pacific, Atlantic and Indian oceans. But I'm going to allow that, to make the seas as small as possible.

In total, this makes $1.29 × 10^{21}$ L of ocean. That's more than the $10^{18}$ m³ I gave earlier, but that was rounded. The actual number given is $1.332 × 10^{21}$ L. So there are $0.042 × 10^{21} ≈ 4 × 10^{19}$ L of seas in the world. This is actually pretty close to my initial guess for the oceans. Still no red.

This does not mean you are safe from the Elder Gods.

Posted on 09 August 2014

Tagged: silly; math