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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 Start Fraction 5 pi over 12 End Fraction radians?

An image shows a circle representing a carousel.An image shows a circle with a horizontal radius labeled 20 feet. The rightmost point on the circle is labeled as the starting point. The ending point is located on the circle in the first quadrant after a rotation of start fraction 5 pi over 12 end fraction radians from the starting point. The ending point is labeled left-parenthesis 20 cosine theta comma 20 sine theta right-parenthesis.
(1 point)
Responses

left-parenthesis 5 left-parenthesis Start Root 2 End Root minus Start Root 6 End Root right-parenthesis comma 5 left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis right-parenthesis
Image with alt text: left-parenthesis 5 left-parenthesis Start Root 2 End Root minus Start Root 6 End Root right-parenthesis comma 5 left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis right-parenthesis

left-parenthesis 5 left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis comma 5 left-parenthesis negative Start Root 2 End Root plus Start Root 6 End Root right-parenthesis right-parenthesis
Image with alt text: left-parenthesis 5 left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis comma 5 left-parenthesis negative Start Root 2 End Root plus Start Root 6 End Root right-parenthesis right-parenthesis

left-parenthesis 5 left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis comma 5 left-parenthesis Start Root 2 End Root minus Start Root 6 End Root right-parenthesis right-parenthesis
Image with alt text: left-parenthesis 5 left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis comma 5 left-parenthesis Start Root 2 End Root minus Start Root 6 End Root right-parenthesis right-parenthesis

left-parenthesis 5 left-parenthesis negative Start Root 2 End Root minus Start Root 6 End Root right-parenthesis comma 5 left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis right-parenthesis
Image with alt text: left-parenthesis 5 left-parenthesis negative Start Root 2 End Root minus Start Root 6 End Root right-parenthesis comma 5 left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis right-parenthesis
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Answer 1

To find the exact position of the rider after the carousel rotates \( \frac{5\pi}{12} \) radians, we can use the parametric equations for the position on a circle based on the angle of rotation.

The position \((x, y)\) of a point on the edge of a circle with radius \( r \) (in this case, \( r = 20 \) feet) can be expressed in terms of the angle \( \theta \) in radians as follows:

\[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \]

Substituting \( r = 20 \) and \( \theta = \frac{5\pi}{12} \):

1. Calculate \( x \): \[ x = 20 \cos\left(\frac{5\pi}{12}\right) \]

2. Calculate \( y \): \[ y = 20 \sin\left(\frac{5\pi}{12}\right) \]

To evaluate \( \cos\left(\frac{5\pi}{12}\right) \) and \( \sin\left(\frac{5\pi}{12}\right) \), we can use the angle sum identities since \( \frac{5\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4} \).

Using trigonometric identities: \[ \cos\left(\frac{5\pi}{12}\right) = \cos\left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \cos\left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{3}\right) \sin\left(\frac{\pi}{4}\right) \] \[ = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) \] \[ = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = \frac{\sqrt{2} - \sqrt{6}}{4} \]

For \( \sin\left(\frac{5\pi}{12}\right) \): \[ \sin\left(\frac{5\pi}{12}\right) = \sin\left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{4}\right) \] \[ = \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) \] \[ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \]

Now substituting back into the parametric equations:

1. Calculate \( x \): \[ x = 20 \left(\frac{\sqrt{2} - \sqrt{6}}{4}\right) = 5(\sqrt{2} - \sqrt{6}) \]

2. Calculate \( y \): \[ y = 20 \left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) = 5(\sqrt{6} + \sqrt{2}) \]

Thus, the coordinate of the rider after rotating \( \frac{5\pi}{12} \) radians is:

\[ \left(5(\sqrt{2} - \sqrt{6}), 5(\sqrt{6} + \sqrt{2})\right) \]

Final answer: \[ \left(5(\sqrt{2} - \sqrt{6}), 5(\sqrt{6} + \sqrt{2})\right) \]

This corresponds to the second option listed:

\[ \left(5(\sqrt{2} + \sqrt{6}), 5(-\sqrt{2} + \sqrt{6})\right) \] So, the correct answer is: left-parenthesis 5 left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis comma 5 left-parenthesis negative Start Root 2 End Root plus Start Root 6 End Root right-parenthesis right-parenthesis.