A man stands on a mountain peak near sunset, with his back to the sun, and sees a rainbow caused by water droplets in air. Since the light is coming horizontally from directly behind him, the redband of the rainbow is visible at an angle of 42° above horizontal from the man's perspective (and the violet band is visible at 40° above horizontal). The droplets causing the rainbow are 7.44 kmaway, along the line of sight to the brightest part of the rainbow. The mountain peak is 2.12 km above a flat plain. What percentage of the complete circular arc of the rainbow is visible to the man?% of a circle

Answered: A man stands on a mountain peak near…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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A man stands on a mountain peak near sunset, with his back to the sun, and sees a rainbow caused by water droplets in air. Since the light is coming horizontally from directly behind him, the red
band of the rainbow is visible at an angle of 42° above horizontal from the man's perspective (and the violet band is visible at 40° above horizontal). The droplets causing the rainbow are 7.44 km
away, along the line of sight to the brightest part of the rainbow. The mountain peak is 2.12 km above a flat plain. What percentage of the complete circular arc of the rainbow is visible to the man?
% of a circle

Expert Solution

Step 1: Given data

From the given data, we know that the most of the intense light located between

$40 degree space a n d space 42 degree$

the man sees a greater percentage of violet inner edge. Then we consider the red outer edge radius of the droplets is

$R space equals 7.44 sin 42 space equals 4.978 k m$

The angle $ϕ$ at which the the rainbow touches the ground is

$cos ϕ space equals fraction numerator 2.12 over denominator R end fraction equals fraction numerator 2.12 over denominator 4.978 end fraction equals 0.426$

Then $ϕ equals cos to the power of negative 1 end exponent open parentheses 0.426 close parentheses$ = 64.78 degrees

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