Rainbows Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation θ of a rainbow is always the same. It can be shown that θ = 4β − 2α. Where
and α = 59.4° and k = 1.33 is the index of refraction of water. Use the given information to find the angle of elevation θ of a rainbow. [Hint: Find sin β, then use the
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Chapter 6 Solutions
Precalculus: Mathematics for Calculus (Standalone Book)
- Vibrating String When a violin string vibrates, the sound produced results from a combination of standing waves that have evenly placed nodes. The figure (1) illustrates some of the possible standing waves. Lets assume that the string has length . a.For fixed t, the string has the shape of a sine curve y=Asinx. Find the appropriate value of for each of the illustrated standing waves. b.Do you notice a pattern in the values of that you found in part a? What would the next two values of be? Sketch rough graphs of the standing waves associated with these new values of . c.Suppose that for fixed t, each point on the string that is not a node vibrates with frequency 440Hz. Find the value of for which an equation of the form y=Acost would model this motion. d.Combine your answers for parts a and c to find functions of the form y(x,t)=Asinxcost that model each of the standing waves in the figure. Assume that A=1. Figure (1)arrow_forwardSurfing the Perfect Wave For a wave to be surfable, it cant break all at once. Robert Guza and Tony Bowen have shown that a wave has a surfable shoulder if it hits the shoreline at an angle given by =sin-112n+1tan where is the angle at which the beach slopes down and where n=0,1,2,. . . . a For =10, find when n=3. b For =15, find when n=2,3, and 4. Explain why the formula does not give a value for when n=0 or 1.arrow_forwardTotal Internal Reflection When light passes from a more dense to a less dense medium-from glass to air, for example-the angle of refraction predicted by Snells Law see exercise 57 can be 90 or large. In this case the light beam is actually reflected Bach into the denser medium. This phenomenon, called total internal reflection, is the principal behind optics. Set 2=90 in Snells Law, and solve for 1 to determine the critical angle of incidence at which total internal reflection begins to occur when light passes from glass to air. Note that the index of refraction from glass to air is reciprocal of the index from air to glass.arrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage