How do I find the horizontal stretch or period of a sinusoidal function if I'm only given the radians per second?

Question

"A Ferris wheel of diameter 16 m rotates at a rate of 0.4 rad/s and is lifted 2m off the ground. If a platform is made so passengers can board at the centre height of the ferris wheel as it goes up,

How do I use 0.4rad/s to determine the period of equation? My teacher never specified rounding so I am thinking it might be an exact answer.

I have tried 2pi/0.4rad, and the v=rw equation. This gives a "k" value in "asin(k(x-d)+c" of approx. 15.708...Is this the correct approach?

Answer 1

I think if you leave the answer in terms of π, it will make more sense.

Period, P = 2π rad/(0.4 rad/s) = 5π seconds

If you need the values of a, k, d and c.

c = radius + 2 = 10 meters

k = 2π/period = 0.4

The range of sine is [-1, +1], so 'a' would be the value of the radius, or 8.

For the phase, d, we need 8sin(0.4(x-d))+10 = 10 when x = 0, so d = 0.

y = 8sin(5πx)+10

Checking,

when x = π/2, y = 18

when x = π, y = 10

when x = 3π/2, y = -8+10 = 2

when x = 2π, y = 10