Cover the Earth

 

The power of advertising – I think most of us would agree, many of our earliest memories involve some sort of advertising. Long before I had any inkling what valves (intake and exhaust, in an IC engine) were, or how important it was to keep them clean, I knew how good STP motor oil was at it. Likewise, from an early age I knew what a powerful tool women wielded, manipulating their husbands’ behavior through judicious selection of coffee brand.

The above pic, or rather an older version thereof, was painted on the wall of a building in Sault Ste Marie, MI, in the 1960s. How long it had been there, and when was it painted over, no idea, but it was there during my early childhood. The building was next to that of my grandmother’s apartment, where mom would drop us kids off in order to do laundry and shopping every week. So, I saw it frequently, and had various “engineering” opinions about it.

Mind you, I knew it was fantasy, and completely unrealistic in terms of physical possibility, not to mention that all life, and pretty much everything else of value on Earth would be destroyed, were the paint company to succeed in carrying out its diabolical scheme. And of course, I comprehended that the idea was to sell enough paint to “cover the earth,” or something like that.

The mid-20^th century, those of you who were around then may attest, was a time of big ideas, and preposterous measuring systems to articulate them. Enough of this could reach the moon and back, enough of those, lined up end to end, could go around the world umpteen times, etc. etc. So the idea of an equivalent amount of paint to “Cover the Earth” was not too far afield.

The other thing that my hyper-analytical young mind stuck on was the size of the can relative to the Earth. I knew that in the graphic, that amount of paint would be far more than you actually needed for the job, or alternatively, if you were to dump that amount of paint on the earth, it would be so thick that it would take the better part of forever to dry.

So, for today’s installment, let’s delve into 3 important questions:

1)     How much paint would be needed to cover the Earth?

2)     How big would the can have to be?

3)     If you dumped a can of paint as large as that in the graphic, how deep would the ocean of paint be?

First, a few practical considerations: The surface of the Earth is not flat, although it’s flatter than you’d think – Mt. Everest and Death Valley notwithstanding, it’s about as smooth as a lightly-used billiard ball. Also, the Earth is not a perfect sphere – it’s a little wider at the equator. & despite the overall smoothness, there are quite a lot of geological features everywhere, not to mention plant life and buildings, roads, bridges, etc. which would consume a lot of paint in their own right. So, let’s do what most engineers and scientists do when confronted with an impossibly complex problem – simplify it through wildly unrealistic assumptions:

1)     The Earth is a perfectly smooth ball

2)     Its surface is drywall, as that’s Sherwin Williams’ main target

3)     Flat vs. spherical – we’ll just ignore curvature

Feel better? I sure do.

Home improvement contractors estimate that a gallon of house paint covers about 400 square feet for a single coat. If you want more detail, you’ll have to ask the exquisite wife of Bruck (EWOB) – she does all the painting. She has rather exacting standards, and through early experience in that endeavor, knows that I don’t. I’ve been accused of strategic incompetence, to which I say au contraire, it’s just good old fashioned incompetence. I am good at a number of things, some of them useful, but painting is decidedly outside of my Venn Diagram.

Let’s say we’re not going to settle for one coat, you need two (that much I do know). So, a gallon will be good for 200 square feet, commensurate with EWOB standards. So, how many square feet are on the surface of an Earth-sized sphere? Geometry!

The radius (r) of the Earth is ~3959 miles (we’re sticking with English units – ‘Murica!).

The surface area of a sphere is 4*pi*r^2 (I’m not using formal mathematical notation because blogspot will almost certainly mangle it).

Thus, the surface area of our drywall sphere, in square miles, is 196,961,284 and change. Converted to square feet (5280 feet in a mile; multiply by 5280^2), it’s (warning: really large number coming!) 5.490965469E15. For those of you who slept through high school math, the E15 means times 10 to the 15^th power. So, five and a half quadrillion, more or less.

In mid-20^th century terms, if square feet were dollar bills, you could make four stacks of them from the Earth to the sun. Of course they’d burn up once you got about halfway there, so I’m not suggesting you actually try this. Side question – when you’re in outer space, which way is up?

Getting back to our little painting project, at 200 square feet per gallon, we’d need 27.45482735E12 of them, so, ~27 and a half trillion. That’s over 41 million Olympic swimming pools! 2 and a half trillion tanker trucks! Equivalent to the amount of water that flows through Niagara Falls for over 14 years! And many more absurd comparisons!

First question answered, onto the next one, i.e., how big of a can would you need? Again, we’re going to make a few simplifying assumptions, namely:

1)     The can has the same relative dimensions as a standard paint can, which are 6.6” diameter (3.3” radius) by 7.5” height. This yields a dimension ratio of r=0.44h, the importance of which will be evident later.

2)     The dimensions of the actual can (thickness, etc.) are negligible.

3)     We’re also going to ignore the fact that paint cans aren’t completely full – they leave a little room for stirring.

That is, we’re going to see how big of a column segment, of the above relative dimensions, would contain 27.5 trillion gallons.

To do that, first we need to convert gallons to cubic linear measurements: 231 in^3 / gal, divided by 12^3 in^3 per cubic foot, yields 3.67E12 cubic feet.

The volume (v) of a column segment is pi*r^2*h, and from the relative dimensions above, that would be pi*(0.44h)^2*h, so h^3=v/(pi*0.44^2), or h=(v/(pi*0.44^2))^(1/3), which resolves to h=18,205.8’, or 3.45 miles. So, a paint can 3.45 miles in height would contain enough paint to cover the earth with two coats. And to complete the visual, the can would be a little over 3 miles wide.

For you east siders, that means such a paint can would cover most of the area between 13 Mile and Big Beaver, from Dequindre to Van Dyke. If you prefer a west side perspective, the area would be between 12 Mile and Maple, from Farmington to Inkster. For those of you not from the suburbs of Detroit, picture any 3-mile square, and how long it would take to circumnavigate it, seeing just about nothing but paint can on one side of you.

Pretty danged large, no? But then picture that against a map of the world, or even North America. Now it seems downright puny. How could that little amount of paint be enough? A huge can to be sure, but it’s still way shorter than Mt. Everest, and you could put it in the Mariana Trench with plenty of room to spare (slowly – we don’t want any tsunamis in SE Asia). How could a can of paint, which would appear as a tiny speck against the backdrop of the entire planet, be able to cover the same, twice?

The answer is our perception of thickness, or rather, relative thickness. Picture yourself placing a gallon of paint in front of a 10’x20’ wall, emptying its contents thereupon in two even coats, and then giving it a good kick down the road (the last part is optional but an important qualification for civil service). When dry, the coating will be about 3 mils (0.003”). Then mentally scale the can size up to a larger wall, then a larger one, etc., until you have a can that would cover the earth. It’d probably be much bigger than 3.5 miles tall (although still not quite the size of the one in the Sherwin Williams ad). Why the disparity? Because unlike the other dimensions, thickness does not scale. Whether you’re painting the spare room or a drywall planet, it’s still 3 mils, which “shrinks” relative to the other dimensions. That probably didn’t help much, did it? Okay then, just trust the math.

Onto question 3, how deep would the deluge be, should a higher power dump upon us the contents of a paint can the size of that in the advertising graphic?

The height of the can looks to be about 1/3 the diameter of the Earth, so, about 2639 miles. Assuming the above relative dimensions, the radius would be about 1161 miles. Recall v=h*pi*r^2, so its volume is about 11.1 billion cubic miles, or ~1.645E21 cubic feet, or ~2.84E24 cubic inches, or ~12.31E21 gallons.

Let’s stick with miles number to compute the depth of the paint deluge. 11.1 billion cubic miles divided by 197 million square miles (note, this is where the importance of ignoring curvature comes into play) yields ~57 miles.

That’s pretty deep, and would easily cover all relief on the Earth’s surface. To picture what this would actually look like, I would first advise, don’t be misled by the graphic. The amount of paint splashing off the North Pole, cascading thickly over the northern hemisphere and equatorial regions, and then falling off in huge droplets below the Tropic of Capricorn (note, the artist was playing fast and loose with gravity here), would require far more paint than that little container could possibly hold. Were you to behold a good-sized classroom globe, about 18” in diameter, 57 miles would add about an eighth of an inch.

Anyway, there’s your answer, 57 miles is the depth of the theoretical ocean of Sherwin Williams paint.

And the final question, not mentioned above, but I’m sure by now it’s burning a hole in your prefrontal cortex: throughout its history, has Sherwin Williams actually sold enough paint to Cover the Earth?

According to cypaint.com, ~16 billion gallons of paint are sold worldwide per year. And per various annual reports, Sherwin Williams, a company that’s been in business since 1866, has a market share of approx. 8.5% in recent years, so, we can presume that they’ve been selling about 1.36 billion gallons of paint per year recently. Disclaimer – don’t hold me to these numbers; just piecing them together from random bits and pieces of difficult-to-find information. In order to estimate the company’s historical paint sales, I’m going to make the wildly irresponsible assumptions that (1) its market share remained constant throughout its history, and (2) total worldwide paint sales has been consistently proportional to population. I realize that this is exceedingly generous, but at least it provides an upper limit. I won’t trouble you with the math this time, but the grand total, under these assumptions, would be about 82 billion gallons, a far cry from the 27.5 trillion needed.

THEREFORE:

1)     We’re all gonna die!

2)     But happily, not by drowning in paint!