Question

For the following exercises, use this scenario: 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 counter clockwise.

When will the child have coordinates (−0.866,−0.5) if the ride last 6 minutes?

Answer 1

Consider the scenario referred in the textbook. So, the child is at an angle of θ = π/2 initially. The carousel revolves counterclockwise. At any particular time, the co-ordinates of the child are (-0.866, -0.5). As the x-coordinate and y-coordinate both are negative, the child will be in the third quadrant. Compute the measurement of the angle as follows:

 

Consider the sine and cosine function in a unit circle,

x = cosθ

x = sinθ

 

Substitute the values of sine and cosine functions as 0.866 and 0.5, compute the reference angles,

cosθ = 0.866

         = cos(π/6)

sinθ = 0.5

         = sin(π/6)

 

The child is in the third quadrant; compute the required angle as follows:

π/6 + π = 7π/6

 

The carousel revolves counterclockwise. So, the measurement of the angle by which the carousel has revolved will be,

7π/6 – π/2 = 7π/6

 

As it will be revolved by an angle of 2π in 1 hour or 60 minutes, the time taken by the carousel to revolved by an angle of 4π/6 will be,

60 × 1/2π × 4π/6 = 20 seconds

 

The ride lasts for 6 minutes. So, it takes 6 rounds. The child will be at the same position after every minute. In first round, the child was at point (-0.866, -0.5) after 20 seconds. In other 5 rounds, he will be at the same point after each minute and 20 seconds.

 

Therefore, the child will be at point (-0.866, -0.5) after,

20 sec, 1min 20 sec, 2min 20 sec, 3min 20sec, 4min 20 sec and 5min 20 sec.

Therefore, the child will be at point (-0.866, -0.5) after,

20 sec, 1min 20 sec, 2min 20 sec, 3min 20sec, 4min 20 sec and 5min 20 sec.