Question

In: Math

What is the coordinates of the child after 125 seconds? For the above 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,..

What is the coordinates of the child after 125 seconds?

For the above 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.

Solutions

Expert Solution

Consider the scenario referred in the textbook. So, the child is at an angle of θ = π/2 initially. The carousel revolves counterclockwise. As it will be revolved by an angle of 2π in 1 hour or 60 minutes, the measurement of the angle by which the carousel revolves in 125 seconds will be,

2π × 1/60 × 125 = 25π/6

 

So, the carousel will be at an angle of π/2 + 25π/6 = 28π/6 after 120 seconds.

 

Consider the sine and cosine function in an unit circle,

x = cosθ

y = sinθ

 

Compute the reference angle of 28π/6 as follows:

28π/6 – 2π = 16π/6

16π/6 – 2π = 4π/6

                    = 2π/3

    π – 2π/3 = π/3

 

So, the values of sine and cosine functions at an angle of 28π/6 will be equal to their values at angle of 2π/3. 

 

The angle is in second quadrant. Compute the values of sine and cosine function as π/3.

For angle π/3, the values will be,

cos(π/3) = 1/2

 sin(π/3) = √3/2

 

As the angle 2π/3 is in second quadrant, the x-coordinate of the required point will be negative and y-coordinate will be positive.

 

Therefore, the co-ordinates of the child after 125 seconds will be (-1/2, √3/2).


Therefore, the co-ordinates of the child after 125 seconds will be (-1/2, √3/2).

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