A ferris wheel is 10 meters in diameter and boarded from a platform that is 3 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 2 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn.

What is the Amplitude?  meters
What is the Midline? y =  meters
What is the Period?  minutes
How High are you off of the ground after 1 minutes? meters

Let's break this down systematically.

### **1. Amplitude**  
The **amplitude** is half the diameter of the ferris wheel, which tells us the distance from the midline to the peak or trough of the wheel's motion.  
- **Diameter** = 10 meters  
- **Amplitude** \(A\) = \( \frac{10}{2} = 5 \) meters  

**Answer: Amplitude = 5 meters**

---

### **2. Midline**  
The **midline** represents the average height of the motion, which corresponds to the center of the ferris wheel plus the height of the boarding platform.  

- The wheel’s center is 5 meters above the platform (half of the diameter).  
- The platform is 3 meters above the ground.  

Thus, the midline is:  
\[
y = 5 + 3 = 8 \, \text{meters}
\]

**Answer: Midline = y = 8 meters**

---

### **3. Period**  
The **period** is the time it takes to complete one full revolution of the ferris wheel. In this case, we are told that the wheel completes one revolution every **2 minutes**.  

**Answer: Period = 2 minutes**

---

### **4. Height after 1 minute**  
We now need to determine the height above the ground after **1 minute**. Since the wheel completes 1 full revolution in 2 minutes, after 1 minute you will be at the **topmost point** of the ferris wheel (halfway through the revolution).

- The amplitude is 5 meters, so the peak is 5 meters above the midline.  
- Since the midline is at 8 meters, the height at the top is:  
  \[
  8 + 5 = 13 \, \text{meters}
  \]

**Answer: Height after 1 minute = 13 meters**

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### **Summary of Answers**  
1. **Amplitude:** 5 meters  
2. **Midline:** \( y = 8 \) meters  
3. **Period:** 2 minutes  
4. **Height after 1 minute:** 13 meters  

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Would you like me to derive the full height function \(h = f(t)\) for this motion? 

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Here are five follow-up questions to deepen your understanding:
1. How would the function change if the ferris wheel completed a revolution in 3 minutes instead?
2. What would be your height at 0.5 minutes (30 seconds)?
3. Can you express the height as a trigonometric function (sine or cosine)?
4. If the platform height were increased to 5 meters, how would that affect the midline?
5. What is the phase shift if the 6 o'clock position no longer corresponds to the starting point?

**Tip:** When modeling with sine or cosine, always think about the amplitude, midline, period, and phase shift to get the full equation.

A ferris wheel is 10 meters in diameter and boarded from a platform that is 3 meters above the ground. The six o'clock position on the ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 2 minutes. The function h = f(t) gives your height in meters above the ground t minutes after the wheel begins to turn. What is the Amplitude? meters What is the Midline? y = meters What is the Period? minutes How High are you off of the ground after 1 minute? meters

Explore the mathematical analysis of a ferris wheel's motion, calculating the amplitude, midline, and period of the height function. Given a ferris wheel with a 10-meter diameter, a 3-meter platform height, and a revolution period of 2 minutes, learn how to find your height above the ground after 1 minute.

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