Precalc Question (answer asap thanks)

Question

A merry-go-round is rotating at the constant angular speed of 3 RPM counterclockwise. The platform of this ride is a circular disc of radius 24 feet. You jump onto the ride at the location pictured below (which is in the link).

https://d2vlcm61l7u1fs.cloudfront.net/media%2Fd79%2Fd79b25e7-76f8-4f02-8917-134f014fb0ae%2Fphpu2xw9v.png

a) If θ = −2.1 rad, then what are your xy-coordinates after 2 hours and 7 seconds?

b) If θ = 2.1 rad, then what are your xycoordinates after 5 seconds?

The answer for a is (-1.674, 23.942) and the answer for b is (23.882, 2.375) but I don't know how to do it

Answer 1

I will help you with a)

The final θ = θ(0) + ωt where ω is the real angular speed in rads/sec and t is time in seconds.

The only difference between a and b will be the input for θ(0) , the starting angle.

Because there are 2π rads per revolution we know that ω = 2πf where f is the frequency in rev/min.

w = 2π(3 rev/min)*(1min/60s = ,31416 rad/seconds (note, I keep all the places of this result on my calculator)

Now we need the time in minutes:

2 hours (3600 s/hr) + 7s = 7207 seconds

Put it all together: θ = -2.1 rad + .314159 rad/s * 7207 sec = 2262.046 rads

Good news: The transformation to (x,y) from (r, θ) is (rcos(θ),rsin(θ))

Remember to put your calculator in radians. You don't need to find the principal angle - the calculator can handle the extra revolutions.

(24cos(2262 rads), 24sin(2262rads) = (23.8822, 2.37486) for answer a, not b!

My answer for b is (-20.720, -12.1103) It seems like the answer key did not have calculator in radians.

Just as a check I approximated things in degrees for (b) 2.1 rads *180 deg/3.14 rad starts at about 120 degrees; in 5 seconds at 3/60 rev/s, you go a 1/4 revolution or 90 degrees. Now at about 210 degrees, principal angle of 30 degrees

so (24* -sqrt(3)/2, 24*(-1/2)) = (-20.8, -12) My values are consistent.

Good luck!