Rainbow Arc Fraction: Solve the Math Problem

[Skynt]

1. A man stands on a mountain peak 2 km from the ground below, and observes a rainbow 8 km away. What fraction of the arc of the rainbow does the man see?

Homework Equations

I tried using a triangle to figure this out but I couldn't come any closer to the answer.
I know that the rainbow largely depends on the rain drop where red has a larger angle of deviation than violet allowing one to see red on top rather than on the bottom. Red is 42 degrees while violet is 40 degrees, etc. etc. The rest of the colors are in between more or less. Understanding this doesn't seem to help me figure it out :(

The Attempt at a Solution

No idea. The teacher gave us the answer, .586 , but that really doesn't help me understand the problem. Please help. Thanks!

 

[tiny-tim]

Hi Skynt!

Don't we need to know the height of the sun?

 

[Skynt]

Well, this problem came from the book - and I agree, that came to my mind as well. The teacher said we didn't need it.

 

[tiny-tim]

Well. unless I'm missing something, the teacher is wrong - see:

http://en.wikipedia.org/wiki/Rainbow​

The position of a rainbow in the sky is always in the opposite direction of the Sun with respect to the observer, and the interior is always slightly brighter than the exterior. The bow is centred on the shadow of the observer's head, or more exactly at the antisolar point (which is below the horizon during the daytime), appearing at an angle of 40°–42° to the line between the observer's head and its shadow. As a result, if the Sun is higher than 42°, then the rainbow is below the horizon and cannot be seen as there are not usually sufficient raindrops between the horizon (that is: eye height) and the ground, to contribute. Exceptions occur when the observer is high above the ground, for example in an aeroplane (see above), on top of a mountain, or above a waterfall.

hmm … the question doesn't say "at a beautiful sunset", does it?