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Trigonometry 2-12 Mathematical Modeling

Trigonometry 2-12 Mathematical Modeling. Ex1. #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 of time. When his stopwatch

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Trigonometry 2-12 Mathematical Modeling

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  1. Trigonometry 2-12 Mathematical Modeling Ex1. #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 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 ten seconds.

  2. at t = 4 sec, d = 16 ft (high point) 18 ft 16 ft

  3. -The period is 10 since the frequency is . a) Sketch a graph of the sinusoid. -The amplitude is 9 since the radius is 9 and the point goes 9 above and 9 below the center. -The axis is 7 since the center of the wheel is 7 feet above the surface of the water. -The phase shift is 4 on cosine, since the point reaches its high point at 4 seconds.

  4. 16 7 t 4 14 -2

  5. b) Write the equation for the sinusoid. A: 9 C: 7 D: 4 c) How far above the surface was the point at 5 seconds, at 17 seconds? d(5) = 14.281 feet d(17) = 4.219 feet

  6. d) When did the point first reach the water’s surface? … t = 0.082 seconds

  7. Ex2. #4 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 at time t = 5.1 years.

  8. a) Sketch the graph of the sinusoid. -The amplitude is 300 since the difference between the max and min is 600. -The period is 4.4 since the difference in time from the min to the next max is 5.1 – 2.9 = 2.2. -The axis is 500, halfway between the max/min. -The phase shift is 5.1 on cosine since the max is reach at t = 5.1 years.

  9. 800 500 200 t 2.9 5.1

  10. b) Write an equation expressing the number of foxes as a function of time. A: 300 C: 500 D: 5.1 c) Predict the population when t = 7 years f(7) = 227 foxes

  11. 800 500 200 t 2.9 5.1 d) Foxes are endangered when their population drops below 300. Between what two values of t were the foxes first endangered? … 300 between t = 2.311 and 3.489 years

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