Writing Sinusoidal Equations

You ride a Ferris Wheel, the height that you are above the ground varies periodically. Consider the height of the center of the wheel to be the equilibrium point. Suppose the diameter of a Ferris Wheel is 42 feet and travels at a rate of 3 revolutions per minute. At the highest point, a seat on the Ferris wheel is 46 feet above the ground. a. Write a sinusoidal equation to model the height of your seat if you got on the Ferris Wheel at its lowest point. b. According to the model, after you have ridden the Ferris Wheel for 4 minutes and 13 seconds, the Ferris Wheel comes to an abrupt stop. What would be the height of your seat at that time?

Let’s look at the height every 0 sec 5 sec 2.5 sec 7.5 sec 10 sec 15 sec 20 sec 12.5 sec 17.5 sec 3 revolutions per min. = = 1 revolution per 20 seconds Let’s look at the height every 2.5 seconds = revolution per 10 seconds = revolution per 5 seconds = revolution per 2.5 seconds

21 -- 46 ft. -- -- 25 ft. -- -- 4 ft. -- 0 sec 5 sec 2.5 sec 7.5 sec Amplitude: 21 Phase Shift: 0 Period: 20 seconds A = -21 C = 0 Vertical Shift: 25 D = 25

21 What is the height at 4 minutes and 13 seconds? -- 46 ft. -- 21 -- 25 ft. -- -- 4 ft. -- 0 sec 5 sec 2.5 sec 7.5 sec 10 sec 15 sec 20 sec 12.5 sec 17.5 sec What is the height at 4 minutes and 13 seconds? 4(60) + 13 = 253 seconds

20 Write the equation of a sine function. (5, 32) (7, 12) (15, 12) (9, -8) Period: 8 Phase Shift: 7 Amplitude: 20 A = -20 Vertical Shift: 12 D = 12

The center of a piston on a windmill tower is 80 meters above the ground. The blades on the windmill are 30 meters in length and turn at a rate of 2 revolutions per minute. The windmill has 3 blades and at the tip of each blade is a light. Two of the lights are white and the 3rd light is blue. We can use a sinusoidal equation to model the height of the light compared to time. If we begin tracking the path of the blue light when the light is at its lowest position at 12:00 midnight (use this as your 0 time), write a sinusoidal equation relating the height to time. Then use your equation to determine the height of the light at 12:10 a.m.

Let’s look at the height every 0 sec 15 sec 3.75 sec 7.5 sec 11.25 sec 18.75 sec 22.5 sec 26.25 sec 30 sec 2 revolutions per min. = = 1 revolution per 30 seconds Let’s look at the height every 3.75 seconds = revolution per 15 seconds = revolution per 7.5 seconds = revolution per 3.75 seconds

30 -- 110 m-- -- 80 m -- -- 50 m- 0 sec 15 sec 30 sec 7.5 sec Amplitude: 30 Phase Shift: 0 Period: 30 seconds A = -30 C = 0 Vertical Shift: 80 D = 80

30 Now, determine the height of the light at 2:10 a.m. -- 110 m-- 0 sec 15 sec 30 sec 7.5 sec 11.25 sec 22.5 sec 3.75 sec 18.75 sec 26.25 sec Now, determine the height of the light at 2:10 a.m.