Posted by **Christine** on Wednesday, February 13, 2013 at 8:30pm.

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 2 minutes. How many minutes of the ride are spent higher than 30 meters above the ground?

- Trig -
**Reiny**, Wednesday, February 13, 2013 at 9:41pm
let's use a sine function ....

amplitude = 20

period = 2π/k = 2

2k = 2π

k = π

so far we have

height = 20 sin π(t + c) + d , assuming we have a phase shift and a vertical shift

clearly, the whole sine curve has to shifted upwards by 24 units, so that the normal min of -20 becomes a min of 4

so we have

height = 20 sin π(t+c) + 24

1. when t = 0 , we want height to be 4

2. when t = 1 , we want the height to be 44

1.

4 = 20sin π(c) + 24

-20 = 20sin πc

sin πc = -1

but we know that sin 3π/2 = -1

then cπ = 3π/2

c = 3/2

equation is

height = 20 sin π(t + 3/2) + 24

check for t = 1, (should get 44)

height = 20 sin π(1 + 3/2) + 24

= 20 sin 5π/2 + 24 = 20(1) + 24 = 44

all looks good.

so we want h ≥ 30

20 sin π(t+3/2) + 24 = 30

sin π(t+3/2) = 6/20 = 3/10 = .3

setting my calculator to radians , I get

π(t+3/2) = .3047 or π-.3047 = 2.837

t + 1.5 = .3047/π or t+1.5 = 2.837/π

t = -1.403 or t = -.597

BUT, we should have positive times,

since the period of the curve is 2, adding 2 to each of these times will also satisfy the equation

so t = -1.403+2 = .597

and t = -.597 + 2 = 1.403

so for the time from t = .597 to t = 1.403 the height will be ≥ 30

So it will be above 30 for( 1.403-.597) minutes

= .806 minutes or appr 48.4 seconds

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