William W. answered • 11/02/19
Experienced Tutor and Retired Engineer
The center of the Ferris Wheel is 18 m above the ground (2 + the 16 m radius). That is the central axis of the graph and so it is the "d" value. The amplitude is 16 (that is the "a" value). If it takes 30 seconds to travel to the top, then it takes 60 seconds to travel all the way around (1 complete cycle). That makes the period = 60 seconds. Using the equation Period = 2π/b then b = 2π/60 = π/30.
So far then the equation is: h(t) = 16cos[π/30(x - c)] + 18 but we need to adjust the c value to model the system as it is. At t = 0, the passengers getting on are at 2 m above the ground. That means the "16cos[π/30(x - c)]" must equal -16 so when we add it to the 18 it gives us 2. To get "16cos[π/30(x - c)]" = -16 then cos[π/30(x - c)] must = -1. At t = 0, it equals 1 so we need to shift it by half a cycle to make it = -1. So c = 30.
Final Equation: h(t) = 16cos[π/30(x - 30)] + 18
For question 2, a sine function exactly matches a cosine function except for the phase shift (x-shift) or c value.instead of shifting by half a cycle, it would be shifted by a quarter cycle or 15 seconds because sin and cos are shifted by 90° or a quarter cycle.
