Applying Trig Functions

Steamboat Problem: Mark Twain sat on the deck of a riverboat Steamboat Problem: Mark Twain sat on the deck of a riverboat. 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 of time. When his stopwatch read 4 seconds, the point was at its highest, 16 feet above the water’s surface. The wheel’s diameter was 18 feet, and it completed a revolution every 10 seconds. Sketch a graph of the sinusoid.

b) Use the graph to write an equation for the sinusoid. c) How far above the surface was the point when Mark’s stopwatch read 5 seconds?, 17 seconds? d) What is the first positive value of time at which the point was at the water’s surface? Was the paddle going in or coming out of the water? Explain your answer. This is quite a problem to begin with but it gives you an idea of the type of problems to come. Next you will do a couple that are not quite as involved.

wheel at a constant speed of radians per second. Jantje is checking her bicycle on a stand. The bicycle’s wheel has a radius of 37 cm and it’s center is 61 cm above the floor. Jantje is rotating the wheel at a constant speed of radians per second. a. Describe the function h(t) using the sine function with h in centimeters and t as time in seconds. b. Find h (t) when t = 9.5 seconds. Round your answer to the nearest tenth cm. c. To the nearest tenth of a second, how long does it take for the wheel to make one revolution?

a.) b.) 2.105-1.688 ≈ 0.417 minutes

4. Tarzan is swinging back and forth on his grapevine. As he swings he goes back and forth across the river bank, going alternately over land and over water.

Jane decides to model his motion mathematically and starts her stopwatch. Let t be the number of seconds the stopwatch reads and let y be the number of meters Tarzan is from the river bank. Assume that y varies sinusoidally with t and that y is positive when Tarzan is over water and negative when he is over land. Jane finds that when t = 2, Tarzan is at one end of his swing, where y is -23. She finds that when t = 5 he reaches the other end of his swing and y = 17.

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