A person gets on a Ferris wheel at the starting point. The starting point is 15 feet off the ground. The Ferris wheel has a radius of 50 feet. What is the height of the rider from the ground after the Ferris wheel rotates Start Fraction 11 pi over 12 End Fraction radians?

Question

An image shows a Ferris wheel.The Ferris wheel is represented by a circle with a dashed horizontal radius. A vertical radius is drawn from the center of the circle to the lowest point on the circle and is labeled 50 feet. The ground below the Ferris wheel is represented by a horizontal line below the circle. A dashed segment goes from the lowest point on the circle to the ground below and is labeled 15 feet. The position of the rider is represented by a point on the circle in the first quadrant labeled left-parenthesis 50 cosine theta comma 50 sine theta right-parenthesis, after a rotation of start fraction 11 pi over 12 end fraction radians from the starting point at the bottom of the Ferris wheel. A dashed perpendicular line goes from the point representing the position of the rider down to the horizontal radius.
(1 point)
Responses

Start Fraction 25 over 2 End Fraction left-parenthesis negative Start Root 2 End Root plus Start Root 6 End Root right-parenthesis plus 65
Image with alt text: Start Fraction 25 over 2 End Fraction left-parenthesis negative Start Root 2 End Root plus Start Root 6 End Root right-parenthesis plus 65

Start Fraction 25 over 2 End Fraction left-parenthesis negative Start Root 2 End Root plus Start Root 6 End Root right-parenthesis
Image with alt text: Start Fraction 25 over 2 End Fraction left-parenthesis negative Start Root 2 End Root plus Start Root 6 End Root right-parenthesis

Start Fraction 25 over 2 End Fraction left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis
Image with alt text: Start Fraction 25 over 2 End Fraction left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis

Start Fraction 25 over 2 End Fraction left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis plus 65
Image with alt text: Start Fraction 25 over 2 End Fraction left-parenthesis Start Root 2 End Root plus Start Root 6 End Root right-parenthesis plus 65
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Answer 1

To find the height of the rider on the Ferris wheel after it has rotated \( \frac{11\pi}{12} \) radians, we can follow these steps:

1. Determine the center of the Ferris wheel: The radius of the Ferris wheel is 50 feet, and the starting point of the rider is 15 feet from the ground. This means that the center of the Ferris wheel is: \[ 15\text{ feet} + 50\text{ feet} = 65\text{ feet} \]

2. Position of the rider: The position of the rider projected onto a circle (considering the center of the Ferris wheel) can be described using the cosine and sine functions. The equations for the coordinates of the rider after rotating an angle \( \theta \) radians are given by: \[ x = \text{center}_x + r \cos(\theta) \] \[ y = \text{center}_y + r \sin(\theta) \]

   Here, \( \text{center}_x \) = 0 (assuming the center is vertically above the starting point), \( \text{center}_y = 65 \text{ feet} \), and \( r = 50 \text{ feet} \). For our case, we use: \[ \theta = \frac{11\pi}{12} \] Thus the equations become: \[ x = 0 + 50 \cos\left(\frac{11\pi}{12}\right) \] \[ y = 65 + 50 \sin\left(\frac{11\pi}{12}\right) \]

3. Calculate the sine and cosine: Using trigonometric values:

   • \( \cos\left(\frac{11\pi}{12}\right) = -\sin\left(\frac{\pi}{12}\right) \)
   • \( \sin\left(\frac{11\pi}{12}\right) = \cos\left(\frac{\pi}{12}\right) \)

   The corresponding values can be expressed using approximate values or rational expressions, but for our purposes: \[ \sin\left(\frac{\pi}{12}\right) \approx \frac{\sqrt{6} - \sqrt{2}}{4} \quad \text{and} \quad \cos\left(\frac{\pi}{12}\right) \approx \frac{\sqrt{6} + \sqrt{2}}{4} \]

4. Insert values: \[ y = 65 + 50 \left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) = 65 + \frac{50(\sqrt{6} + \sqrt{2})}{4} \] \[ y = 65 + \frac{25(\sqrt{6} + \sqrt{2})}{2} \]

5. Final expression: Thus, the height of the rider from the ground after \( \frac{11\pi}{12} \) radians is: \[ y = 65 + \frac{25}{2} \left(\sqrt{6} + \sqrt{2}\right) \]

Looking at the provided responses, the answer must match one of those. Comparing it with the options given:

Final Answer:

Thus, the height of the rider from the ground after a rotation of \( \frac{11\pi}{12} \) radians is: \[ \frac{25}{2}(\sqrt{2} + \sqrt{6}) + 65 \]