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Math Analysis AB Using Sine and Cosine Functions as Models Worksheet 2.2 Warm Up Find the amplitude, period, and phase shift for the sine function whose graph is shown. Write an equation for this graph. Now try finding a cosine function with this same graph. MATHEMATICAL MODELING Sine and cosine functions can be used to model many real-life situations including electric currents, musical tones, radio waves, tides, sunrises, and weather patterns. Example: Throughout the day, the depth of water at the end of a dock varies with the tides. The table shows the depth (in meters) at various times during the morning. t (time) Midnight 2 a.m. 4 a.m. 6 a.m. 8 a.m. 10 a.m. Noon y (depth) 2.55 3.80 4.40 3.80 2.55 1.80 2.27 a. Use a trigonometric function to model this data. b. Find the depth at 9 a.m. and 3 p.m. c. A boat needs at least 3 meters of water to moor at the dock. During what times in the afternoon can it safely dock? Math Analysis AB Modeling Problems 1. Household electrical power in the US is provided in the form of alternating current. Typically the voltage cycles smoothly between +155.6 and -155.6 volts 60 times per second. Use a cosine function to model the alternating voltage. 2. A rabbit population in a national park rises and falls each year. It is at its minimum of 5000 rabbits in January. By July, as the weather warms up and food grows more abundant, the population triples in size. By the following January, the population again falls to 5000 rabbits, completing the annual cycle. Us a trigonometric function to find a possible formula for R = f(t) where R is the size of the rabbit population as a function of t, the number of months since January.