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Unread 14-12-2012, 20:34
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Richard Wallace Richard Wallace is offline
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Re: numerical computation contest

Russ,

I understand you intended this one as a programming exercise -- hence you posted it in the programming forum.

However, it can also serve as an example of engineering approximation. Jon showed earlier how the Pythagorean theorem gives 51.48 ft as a rough approximation.

For a slightly better one, I used the Taylor series expansion for the sinc function, defined as sinc(x) = sin(x)/x. The first two terms of that expansion are:

sinc(x) = 1 - (x^2)/6 + ....

The arc length is 2rx and the chord length is 2rsin(x), where 2x is the angle subtended by the arc. Since the chord is one mile long and the arc is one foot longer, we have:

sinc(x) = 5280/5281

and the truncated series approximation above gives

x = sqrt(6/5281) = 0.03371 radians, or about 1.931 degrees.

Then the arc is part of a circle with perimeter

(5281/5280) x (180/1.931) = 93.23 miles,

so the radius r is 14.84 miles [this is a very flat arc!]

and the height h is 14.84 x (1 - cos(x)) miles = 44.49 ft.

Back in the day, engineers without benefit of cheap computers could have gotten this close using slide rules; i.e., four figures.
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Richard Wallace

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I believe in intuition and inspiration. Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.
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Last edited by Richard Wallace : 14-12-2012 at 20:49.