A Ferris wheel is 30 meters in diameter and is raised on a platform 8 feet from the ground so that the bottom of the wheel is in line with the top of the platform. A ride lasts 20 minutes and undergoes five revolutions in a clockwise direction during this time. If Brad is sitting in a pod in position *A* at time *t* = 0 minutes, determine his position when he reaches *B* and *E*.

*B*: (1, 23) and *E*: (4, 38) *B*: (4, 23) and *E*: (16, 38) *B*: (1, 53) and *E*: (4, 68) *B*: (4, 53) and *E*: (16, 68)

To determine Brad's position on the Ferris wheel at points *B* and *E*, we need to analyze the movement of the wheel.

### Key Information
- **Diameter of the Ferris wheel:** 30 meters (radius = 15 meters)
- **Height of the platform:** 8 feet (approximately 2.44 meters)
- **Total ride time:** 20 minutes
- **Total revolutions:** 5 revolutions

### Calculating Angular Speed
1. **Total time of ride in seconds:**
   \[
   20 \text{ minutes} = 20 \times 60 = 1200 \text{ seconds}
   \]
2. **Angular speed (in radians per second):**
   \[
   \text{Total angle} = 5 \times 2\pi = 10\pi \text{ radians}
   \]
   \[
   \text{Angular speed} = \frac{10\pi}{1200} = \frac{\pi}{120} \text{ radians/second}
   \]

### Position Calculation
Brad's position can be modeled using parametric equations based on the angle of rotation \(\theta\):

- **Vertical position (height):**
  \[
  y(t) = 2.44 + 15(1 - \cos(\theta(t)))
  \]
- **Horizontal position:** (not necessary for this problem, but can be given by)
  \[
  x(t) = 15\sin(\theta(t))
  \]

### Time Calculation for Points
To find the positions at points *B* and *E*:

1. **For *B* and *E***:
   - Convert the minutes into seconds for calculations.
   - Calculate the angle at each time by using:
   \[
   \theta(t) = \frac{\pi}{120} t
   \]
   - Plug \(t\) values into the \(y(t)\) equation to find the height at those times.

#### For *B*:
- **At \(t = 1\) minute = 60 seconds:**
  \[
  \theta(60) = \frac{\pi}{120} \times 60 = \frac{\pi}{2}
  \]
  \[
  y(60) = 2.44 + 15(1 - \cos(\frac{\pi}{2})) = 2.44 + 15(1 - 0) = 17.44 \text{ meters}
  \]

- **At \(t = 4\) minutes = 240 seconds:**
  \[
  \theta(240) = \frac{\pi}{120} \times 240 = 2\pi
  \]
  \[
  y(240) = 2.44 + 15(1 - \cos(2\pi)) = 2.44 + 15(1 - 1) = 2.44 \text{ meters}
  \]

#### For *E*:
- **At \(t = 16\) minutes = 960 seconds:**
  \[
  \theta(960) = \frac{\pi}{120} \times 960 = 8\pi
  \]
  \[
  y(960) = 2.44 + 15(1 - \cos(8\pi)) = 2.44 + 15(1 - 1) = 2.44 \text{ meters}
  \]

### Final Positions
- For *B*: 
  - At \(t = 1\): (not matching options)
  - At \(t = 4\): (2.44 meters)

- For *E*:
  - At \(t = 4\): (2.44 meters)
  - At \(t = 16\): (2.44 meters)

### Conclusion
Brad's positions at the specified times do not match the given options directly based on the calculations. We need to check the rounding or other context-based adjustments.

If you want further clarification or additional calculations, please let me know! Here are some related questions to consider:
1. How does the height of the Ferris wheel affect the calculation?
2. What would the positions be if the wheel spun counterclockwise?
3. How can you represent the positions on a graph?
4. What adjustments would you make for a different diameter?
5. How does the angular speed influence the overall experience of the ride?

**Tip:** Always double-check calculations to ensure accuracy, especially in applications involving periodic functions.

A Ferris wheel is 30 meters in diameter and is raised on a platform 8 feet from the ground so that the bottom of the wheel is in line with the top of the platform. A ride lasts 20 minutes and undergoes five revolutions in a clockwise direction during this time. If Brad is sitting in a pod in position *A* at time *t* = 0 minutes, determine his position when he reaches *B* and *E*.

Explore the mechanics of a Ferris wheel ride with a 30-meter diameter and an 8-foot platform. This detailed solution covers the calculation of Brad's position at specific times during his ride, including the use of angular speed and parametric equations. Ideal for high school students studying trigonometry and circular motion.

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