Question

A person is on the outer edge of a carousel with a radius of 20 feet that is rotating counterclockwise around a point that is centered at the origin. What is the exact value of the position of the rider after the carousel rotates 5pi/12 radians?

Answer 1

The exact position of the rider after the carousel rotates 5pi/12 radians is (-10 / √(6), 10 / √(3)).

Step-by-step explanation:

To find the exact value of the position of the rider after the carousel rotates 5pi/12 radians, we need to use the formula for the position of a point on a circle. The formula is x = r * cos(theta) and y = r * sin(theta), where r is the radius of the circle and theta is the angle in radians. In this case, the radius of the carousel is 20 feet. Plugging in the given angle of 5pi/12 radians, we can calculate the exact position of the rider.

Let's start by finding the x-coordinate: x = 20 * cos(5pi/12) = -20 / (2 * √(6)) = -10 / √(6).

Next, let's find the y-coordinate: y = 20 * sin(5pi/12) = 20 / (2 * √(3)) = 10 / √(3).

Therefore, the exact position of the rider after the carousel rotates 5pi/12 radians is (-10 / √(6), 10 / √(3)).

Answer 2

Answer: Assuming the riders starts at the position (20, 0) on the x-axis, the exact position of the rider will be (20cos75, 20sin75) or about (5.18, 19.32).

The angle for 5pi/12 radians is 75 degrees. Therefore, to find the position we can use the sine and cosine of 75 to find the x and y value of the coordinate.

For the y-value, we can write and solve:
sin75 = x/20

For the x-value, we can write and solve:
cos75 = x/20