When I was walking to my office today, I was struck with this potentially interesting applied math problem.

I was looking at the sky, due east, I saw a fat and juicy rain cloud hanging at an angle of about 60 degrees to the horizon. The question that came to my mind was:

"How far away is the place where this particular rain cloud is directly over-head?"

To make the problem tractable, let us assume (i) wind effects are negligible, (ii) the earth is a perfect sphere, (iii) the local topography is perfectly flat (no mountains or valleys).

These simplifications allow me to draw a conceptual picture like:

The inner circle represents the earth with radius \(R\). You are at the point

Let us assume that this rain cloud is at a height \(h\) from the surface of the earth, so that the length of the line segment

Clearly, the picture is not to scale!

To summarize, we know \(R, h,\) and \(\theta\). If we can somehow find the angle \(\phi\) then we can infer the length of the chord

I was looking at the sky, due east, I saw a fat and juicy rain cloud hanging at an angle of about 60 degrees to the horizon. The question that came to my mind was:

"How far away is the place where this particular rain cloud is directly over-head?"

**Formulation**To make the problem tractable, let us assume (i) wind effects are negligible, (ii) the earth is a perfect sphere, (iii) the local topography is perfectly flat (no mountains or valleys).

These simplifications allow me to draw a conceptual picture like:

The inner circle represents the earth with radius \(R\). You are at the point

**R**in the picture. You observe the cloud at position**P**, which makes an angle of \(\theta\) with the horizon denoted by the line H1-H2.Let us assume that this rain cloud is at a height \(h\) from the surface of the earth, so that the length of the line segment

**OP**is \(R+h\).Clearly, the picture is not to scale!

**S**is the location where the cloud is directly overhead. We want to find the length of the arc**RS**.To summarize, we know \(R, h,\) and \(\theta\). If we can somehow find the angle \(\phi\) then we can infer the length of the chord

**RS**, which would be the distance we seek.
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