Pre calculus sinusoidal graphs

Question

A ferris wheel is 40 meters in diameter and boarded from a platform that is 4 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 4 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn.

What is the amplitude?

What is the midline?

What is the period?

How high are you off the ground in two minutes?

Answer 1

Use the function f(t) = a·sin(bx+c)+d since we start at the minimum:

Amplitude: |a| --> (40-4)/2=18

Period: 2π/|b| --> 2π/|b| = 4 --> b=π/2

Phase/Horizontal Shift: -c/b = 0 --> -c/(π/2) = 1 --> c = -π/2

Midline: y=d=(40+4)/2=22

Final function is f(t) = 18sin(πx/2+π/2)+22

After t=2 minutes:

f(2) = 18sin(2π/2+π/2)+22 = 18sin(3π/2)+22 = 18(-1)+22 = -18+22 = 4

Hope this helped!

Answer 2

40 meters in diameter means an amplitude = A = 40/2 = 20 meters

height = h(t) = AsinB(t-C) +D

where A = amplitude=20

B = 2pi/period 2pi/(1/4)

C = phase shift

D = midline= vertical shift = 4

period= 1 revolution per every 4 minutes

p = 1/4

B = 2pi/(1/4) = 8pi

D = 4

A = 20

at 6 o'clock h=4

h=4 = 20sin(8pi)(t+C) = 4

sin[8pi(C) = 0

8piC = 0

C=0

h(t) = 20sin[8pi(t)]+ 4

or you could do this with cosine, then there is a phase shift C, not 0