Charles Petzold: Very Wrong Math

The difference between misinformation and disinformation is the difference between ignorance and malice. Trolling is somewhat different, incorporating an element of provocation and narcissism. But what the hell is this?

This was posted by a design and construction company that Facebook thinks I should follow. A cursory glance at their other posts reveals nothing quite as egregiously wrong. So I’m confused. Is it supposed to be a joke?

Of course, me being me, I was curious exactly how wrong it was.

Let me ignore the “flight time” part of the problem and focus on the length of those two arcs. The length of a circular arc is proportional to the subtending angle and the radius of the circle, specifically:

$$\mathrm{length\; of\; arc}=2·\mathrm{\pi }·\frac{\mathrm{angle}}{\mathrm{360°}}·\mathrm{radius}$$

The angle for the two arcs in the illustration is the same, but in calculating the length of those arcs, the radius of the earth has to be taken into account as well as the distance above the surface of the earth. If the radius of the earth is R, then this illustration implies that:

$$R+\mathrm{33,000\prime }=4·(R+\mathrm{5,000\prime })$$

The radius of the earth would therefore be:

$$R=\mathrm{4,333\prime }$$

That’s a distance in feet! That would make the circumference of the earth a bit over 5 miles, which is considerably less than the actual circumference.

The mean radius of the earth is actually 3,459 miles or over 18 million feet. Setting R to that value, the ratio of the length of the outer arc to the inner arc is therefore:

$$\frac{R+\mathrm{33,000\prime }}{R+\mathrm{5,000\prime }}=1.0015$$

In other words, the outer arc is less than 1% longer than the inner arc, but the flight time at that altitude would likely be less because of the decreased air resistance.