Math CentralQuandaries & Queries


Question from grant, a parent:

if you have 2 plumb structures 10ft. tall, how far apart will they be when they are 1 inch out of parallel at the top

We assume that the Earth is a perfect sphere (which it is not) and look at the cross section determined by these two  10 ft  structures:


Now we use radian measure, namely the equation:  θ = s/r

Since these structures are plumb, they lie perfectly on the radial lines of the Earth.  Thus,  θ is the same for the small and big circles, and we obtain:

Therefore the difference between these two arc lengths is,

So we wish to solve for s1 knowing that s2 - s1 = 1 inch = 1/12 foot, and r1 = r2 + 10 feet.

What is the radius of the Earth in feet?  How far apart are the two structures?

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