(1) Trigonometry 8

(1) Trigonometry 8 - Pre Calculus Math 40S Explained...

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Pre – Calculus Math 40S: Explained! www.math40s.com 76 Trigonometry Lesson 8: Part I Ferris Wheels One of the most common application questions for graphing trigonometric functions involves Ferris wheels, since the up and down motion of a rider follows the shape of a sine or cosine graph. Example: A Ferris wheel has a diameter of 30 m, with the centre 18 m above the ground. It makes one complete rotation every 60 s. a) Draw the graph of one complete cycle, assuming the rider starts at the lowest point. b) Find the cosine equation of the graph. c) What is the height of the rider at 52 seconds? 10 20 30 40 50 60 5 10 a. When you look at a Ferris wheel it makes a circular motion. Do NOT draw a circle as your graph. The graph represents the up & down motion over time. In the first half of the graph, we can see the person will go up from 3 m to 33 m in 30 seconds. In the second half, the person will go back down to 3 m. d) At what time(s) is the rider at 20 m? ---------------------------------------------------------------- 35 15 20 25 30 Midline at 18 m Height (m) a-value: We know the diameter of the wheel is 30 m, so the radius (which is the same as amplitude) will be 15 m. b-value: The period is 60 s, so: b. Time (s) c. Method 1: Plug 52 s in for time in either the cosine equation. Simplify and calculate with your TI-83 in radian mode ( ) 15cos 18 30 (52) 15cos 52 18 30 (52) 7.96 m h t t h h π π = − + = − + = Method 2: Put your calculator in radian mode and graph either the cosine function. 2 nd Trace Æ Value Æ x = 52 This will calculate the height directly from the graph. 2 2 60 30 b P π π π = = = c-value: There is none in the cosine pattern. d-value: The centre of the wheel is 18 m above the ground, so that will be the midline The cosine pattern is upside down, so you need to put a negative in front. h(t)= -15cos t+18 30 π d. To find the times the rider is at 20 m, graph a horizontal line in your TI-83 at y = 20, then find the points of intersection.
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