## Monday, June 8, 2015

### Cloudy Puzzle

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?"

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.