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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 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. 18 ft y 4 sec 16 ft (max) One revolution every 10 sec. 10 x 4 Height (ft) 8 12 16 Time (sec) Write equation of the sinusoid. Period: 10 seconds per cycle 2 10 b b 5 Phase shift: Maximum starts at 4 seconds Vertical Shift: middle of graph (16 + -2)/2 = 7 Amplitude: distance from middle of graph 16 – 7 = 9 (no reflection if starting at max) Therefore, the equation is: y 9 cos 5 ( x 4) 7 How far above the surface was the point when Mark’s stopwatch read 5 seconds? 17 seconds? By using the equation: y 9 cos 5 ( x 4) 7 We can plug in 5 and 17 into the x and solve for y (on calculator with trace or table) When x = 5 sec , y = 14.3 ft When x = 17 sec , y = 4.2 ft What is the first positive value of time at which the point was at the water’s surface? Find the intersection of y = 0 (water’s surface) and the sinusoid representing the paddlewheel. Water’s surface