Question

Reid gets on a Ferris Wheel from a platform 3 feet above the ground (which is the low point of the rotation). The diameter of the Ferris Wheel is 20 feet, and one revolution takes 24 seconds. Write a function for Reid's height, in feet, in terms of his time, in seconds. a) Use your answer from problem 28 to determine Reid's height after 1 minute. b)Use your answer from problem 28 to determine when Reid's height will first reach 18 feet.

Answer 1

This corresponds to a sinusoidal graph, we have:

Radius= 20 feet, that corresponds to the amplitud (A)

One revolution takes 24 seconds, that corresponds to the period

Min= 3 feet

k=360/24=15

D= bottom distance+ amplitud=3+20=23 feet

$\begin{gathered} \text{The movement is represented by the expression:} \\ y=A\cos (kx)+D \end{gathered}$

Then, let x be the time, in seconds:

$y=20\cos (15x)+23$

a) To determine Reid's height after 1 minute= 60 seconds, substitute x=60

$\begin{gathered} y=20\cos (15\cdot60)+23 \\ y=9\text{ f}eet \end{gathered}$

b) To determine when Reid's height will first reach 18 feet, we have to substitute y=18 feet, and isolate x:

$\begin{gathered} 18=20\cos (15x)+23 \\ 18-23=20\cos (15x) \\ -5=20\cos (15x) \\ -\frac{5}{20}=\cos (15x) \\ -\frac{1}{4}=\cos (15x) \\ By\text{ trigonometric properties:} \\ 15x=\cos ^{-1}(-\frac{1}{4})+2\pi n \\ x=\frac{\cos^{-1}(-\frac{1}{4})}{15}+\frac{2\pi n}{15} \end{gathered}$