math problem - Trig

Question

A child enters a carousel that takes one minute to revolve once around. The child enters at the point (0, 1),

 that is, on the due north position. Assume the carousel revolves counterclockwise when viewed from above.

What are the coordinates of the child after 150 seconds? (Round your answer to three decimal places.)

Answer 1

Draw a diagram!

Since one full revolution takes one minute, or 60 seconds, and there are four quadrants along the unit circle, 60/4=15 seconds, so it takes 15 seconds to move to the next axis point along the circle

150 seconds = 2 minutes and 30 seconds

Each minute, the child starts at (0,1) and makes one full revolution around the circle, ending back at (0,1)

For the remaining 30 seconds, he moves half the distance he would in one minute, meaning he moves halfway along the circle, ending at (0,-1)

Answer 2

Looking down on the carousel, it would look something like this:

As mentioned, at time t = 0, the coordinate is (0, 1). After 15 seconds, the child would be at (-1, 0), after 30 seconds (total) the child would be at (0, -1), then after 45 seconds the child would be at (1, 0). Finally after 60 seconds (1 minute), the child would be back at (0, 1) and would start repeating this cycle.

Considering ONLY the x-value which have ordered pairs of (time in seconds, x-position) or (t, x) of (0, 0), (15, -1), (30, 0), (45, 1), (60, 0). This is a negative sine wave (starts at zero, goes down to -1, up passed zero to 1 and then back down to zero. The equation, considering that the period is 60 seconds, would be x = -sin[(2π/60)t] or x(t) = -sin[(π/30)t]. where t is in seconds. To find the x-coordinate after 150 seconds, plug in t = 150 to get x(150) = -sin[(π/30)150] = -sin(5π) = 0

Considering ONLY the y-value which have ordered pairs of (time in seconds, y-position) or (t, y) of (0, 1), (15, 0), (30, -1), (45, 0), (60, 1). This is a cosine wave (starts at 1, goes down passed zero to -1 and then back up passed zero to 1. The equation, considering that the period is 60 seconds, would be y = cos[(2π/60)t] or y(t) = cos[(π/30)t]. where t is in seconds. To find the y-coordinate after 150 seconds, plug in t = 150 to get y(150) = cos[(π/30)150] = cos(5π) = -1

So the (x, y) coordinates would be (0, -1)