To solve this problem, we need to understand the position of the rider on the Ferris wheel and how the height changes depending on the angle $\theta$.

Key Information:

• The Ferris wheel's diameter is 310 feet.
• The top of the wheel is 320 feet above the ground.
• $\theta = 120^\circ$, the angle formed as the rider moves.

Step-by-Step Solution:

1. Radius of the Ferris Wheel: Since the diameter of the Ferris wheel is 310 feet, the radius is: $r = \frac{310}{2} = 155 \, \text{feet}$

2. Center of the Ferris Wheel: The top of the wheel is 320 feet above the ground. Since the radius is 155 feet, the center of the Ferris wheel (which is 155 feet below the top) is: $\text{Height of the center} = 320 - 155 = 165 \, \text{feet}$

3. Rider's Position: The rider's vertical position at any angle $\theta$ can be modeled using the cosine function. When the angle $\theta$ is measured from the rightmost position (position $P_0$, directly to the right of the center), the height of the rider above the center is given by: $h_{\text{relative}} = r \cos(\theta)$ This gives the height of the rider relative to the center of the Ferris wheel.

4. Calculating $h_{\text{relative}}$ for $\theta = 120^\circ$: Since $\theta = 120^\circ$, we compute the cosine: $\cos(120^\circ) = -\frac{1}{2}$ Now, calculate the rider's height relative to the center: $h_{\text{relative}} = 155 \times \left( -\frac{1}{2} \right) = -77.5 \, \text{feet}$ This means the rider is 77.5 feet below the center of the Ferris wheel.

5. Total Height Above the Ground: The total height of the rider above the ground is the height of the center plus the rider’s height relative to the center: $h_{\text{total}} = 165 + (-77.5) = 165 - 77.5 = 87.5 \, \text{feet}$

Final Answer:

The rider's height above the ground when $\theta = 120^\circ$ is 87.5 feet.

Would you like more details or have any questions?

Related Questions:

1. What would be the height of the rider when $\theta = 90^\circ$?
2. How would the formula change if the Ferris wheel had a different diameter?
3. What is the rider’s height when $\theta = 0^\circ$ or $\theta = 360^\circ$?
4. How would you calculate the height if the Ferris wheel rotated in the opposite direction?
5. How does the height change as the angle $\theta$ increases from 0 to 360 degrees?

Tip:

For problems involving circular motion, using trigonometric functions (sine or cosine) is a powerful way to model vertical and horizontal movement.