## Finding Rotation Angles: path of merry-go-round horses

Trigonometric ratios and functions, the unit circle, inverse trig functions, identities, trig graphs, etc.
bishop101
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### Finding Rotation Angles: path of merry-go-round horses

Hi, I was wondering if someone could help me with a Advanced functions Trig question that I'm having pr
oblems with.

On a merry-go-round, each horse moves up and down five times in one complete revolution. Imagine that each horse rises and falls 25 cm from its centre position. The up-and-down mothion of each horse can be modelled by the function

h(t) = 25 cos (5Beta) , where h is the horses displacement from its centre and Beta is the rotation angle of the merry-go-round. Assume the ride begins when,
Beta = 0degrees for a given horse.

(Sorry I don't know the signs for Angel Beta, or degrees on the keyboard.)

a) Determine the displacement of one horse at the start of the ride.
b) Ar what rotation angles will the horse be displaced 15 cm in one revolution?
c0 At what rotation angles will the horse be displaced -20cm in one revolution? ( first four angles only.)
d) How long will it take for one complete revolution if the carousel rotates at a speed of 24degrees/s (24 degrees per second.)

thanks.

stapel_eliz
Posts: 1628
Joined: Mon Dec 08, 2008 4:22 pm
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(Sorry I don't know the signs for Angel Beta, or degrees on the keyboard.)
For text-only formatting, simple spell the angle names (or replace them with letters, such as "b" for "beta", if you like). For Greek formatting, use the LaTeX commands: $$\beta[$$ displays $\beta$, etc.
On a merry-go-round, each horse moves up and down five times in one complete revolution. Imagine that each horse rises and falls 25 cm from its centre position. The up-and-down mothion of each horse can be modelled by the function h(t) = 25 cos (5Beta) , where h is the horses displacement from its centre and Beta is the rotation angle of the merry-go-round. Assume the ride begins when Beta = 0degrees for a given horse.

a) Determine the displacement of one horse at the start of the ride.
Plug "0" in for "beta", and evaluate.

. . . . .h(0) = 25 cos(0)

...and so forth.
b) Ar what rotation angles will the horse be displaced 15 cm in one revolution?
Plug "15" in for "h(t)", and solve:

. . . . .15 = 25 cos(5b)

. . . . .3/5 = cos(5b)

. . . . .cos-1(3/5) = 5b

. . . . .(1/5) cos-1(3/5) = b

...and so forth.
c0 At what rotation angles will the horse be displaced -20cm in one revolution? ( first four angles only.)
Plug "-20" in for "h(t)", and solve:

. . . . .-20 = 25 cos(5b)

. . . . .-4/5 = cos(5b)

. . . . .cos-1(-4/5) = 5b

. . . . .(1/5) cos-1(-4/5) = b

...and so forth.
d) How long will it take for one complete revolution if the carousel rotates at a speed of 24degrees/s (24 degrees per second.)
A "revolution" is "once around". The carousel rotates at 24 degrees per second, and there are 360 degrees in one revolution: (360 degrees / revolution)/(24 degrees / second) = 15 seconds / revolution.