Concept explainers
APPLICATIONS
Rainbows Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation
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Algebra and Trigonometry (MindTap Course List)
- 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_forwardHeight of a Mountain To calculate the height h of a mountain, angles and and distance d are measured, as shown in the figure below. a Show that h=dcotcot b Show that h=dsinsinsin() c Use the formulas from parts a and b to find the height of a mountain if =25, =29 and d=800 ft. Do you get the same answer from each formula?arrow_forward
- The designers of a water park have sketched a preliminary drawing of a new slide (see figure). (a) Find the height h of the slide. (b) Find the angle of depression from the top of the slide to the end of the slide at the ground in terms of the horizontal distance d a rider travels. (c) Safety restrictions require the angle of depression to be no less than 25 and no more than 30. Find an interval for how far a rider travels horizontally.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_forwardShadow length The length s of a shadow cast by a vertical gnomon (a device used to tell time) of height h when the angle of the sun above the horizon is can be modeled by the equation s=hsin90sin 090 (a) Verify that the expression for s is equal to h cot . (b) Use a graphing utility to create a table of the lengths s for different values of . Let h=5 feet. (c) Use your table from part (b) to determine the angle of the sun that results in the minimum length of the shadow. (d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is 90?arrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning