Name: Math 4. a. Graph two periods of the function T(t). Helpful Numbers. Min = Max = Midline = Amplitude =

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1 Name: Math 4 February 3/4, 2016 Sinusoidal Application Problems Sinusoidal Application Problems Objective: Practice creating and using sinusoidal function models. 1. The temperature varies sinusoidally on a certain day in May. The minimum temperature is 55 F at midnight. The maximum temperature is 70 F at noon. Let t be the number of hours since midnight (t = 0 at midnight). a. Graph two periods of the function T(t). Helpful Numbers Min = Max = Midline = Amplitude = b. Find an equation for T(t) using the cosine function. Period = c. Find an equation for T(t) using the sine function. d. Use your equation to find the temperature at 10am. e. I want to go on a bike ride on this day. I prefer to ride when the temperature is at least 65 F. What is the earliest time of day that I can leave for my ride? How long can I stay out before I get cold? (Hint: use graphs on your calculator.)

2 page 2 2. Mark Twain sat on the deck of a river steamboat. As the paddlewheel turned, a point on the paddle blade moved in such a way that its distance, d, from the water's surface was a sinusoidal function. When his stopwatch read 5 seconds, the point was at its highest, 16 feet above the water's surface. The wheel's diameter is 18 ft and it completed a revolution every 10 seconds. a. Graph two periods of the function d(t). Helpful Numbers Min = Max = Midline = Amplitude = b. Find the equation of for d(t). Period = c. Use your equation to find the distance from the surface of the water after 28 seconds.

3 page 3 3. In the waters near Hull, MA, the average high tide is 10 feet; that is, the level of water at high tide exceeds the level of water at low tide by 10 feet (low tide would be considered 0 ). The tide comes in and goes out every 24.4 hours. That is, at t = 0 hours, the tide is highest, and at t = 24.4 hours, the tide is highest again. a. Sketch two full revolutions of the graph that models the height of the tide in Hull (ft) vs. time (hours) b. Write an equation of a sinusoid expressing the height of the tide as a function of time, t. c. Find the height of the tide near Hull 5 hours after high tide. (Round to the nearest 10 th.) d. At what time of day will the height of the tide first reach 2 feet? (Use a calculator graph to solve.)

4 page 4 4. Predators: Naturalists find that the populations of some kinds of predatory animals vary periodically. Assume that the population of foxes in a certain forest varies sinusoidally with time. Records started being kept when time t = 0 years. A minimum number, 200 foxes, occurred when t = 2.9 years. The next maximum, 800 foxes, occurred when t = 8.7 years. a. Sketch a graph of the sinusoid b. Write an equation expressing the number of foxes as a function of time, t. c. Predict the population when t = Sunspot Problem: For several hundred years, astronomers have kept track of the number of solar flares, or sunspots, which occur on the surface of the sun. The number of sunspots counted in a given year varies periodically from a minimum of about 10 per year to a maximum of about 110 per year. Between the maximums that occurred in the years 1750 and 1948, there were 18 complete cycles. a. What is the period of the sunspot cycle? b. Assume that the number of sunspots counted in a year varies sinusoidally with the year. Sketch a graph of two sunspot cycles, starting in 1948, thus let 1948 = 0 on the x axis. c. Write an equation expressing the number of sunspots per year in terms of the year d. How many sunspots would you expect in the year 2006?

5 page 5 6. Bicycle Problem As you ride a bicycle, the distance between your foot and the pavement varies sinusoidally with the horizontal distance the bicycle has gone. Suppose that you start with your right foot somewhere between a high point and a low point, and push down. When you have gone 7 m, your right foot first reaches its lowest point, 11 cm above the pavement. The high points are 45 cm above the pavement. The bicycle moves a horizontal distance of 20 m for each complete revolution of the pedals. a. Sketch a graph of the sinusoid. b. Write an equation for the sinusoid using the sine function. c. Write an equation for the sinusoid using the cosine function.