A ferris wheel, with a diameter of 40 m , makes a full rotation in 3 minutes. Passengers board at the
bottom of the ferris wheel, which is 2 m above ground.
a) Find the equation that best models the height of a passenger.
b) Find the height of a passenger who has been riding the ferris wheel for 150 seconds.

Question

UpStudy AI · Accepted Answer

To model the height of a passenger on the Ferris wheel over time, we'll analyze the given information step by step.

### Given:
- **Diameter of Ferris Wheel:** 40 meters
- **Radius (A):** $$ \frac{40}{2} = 20 $$ meters
- **Time for one full rotation (Period, $$ T $$):** 3 minutes = 180 seconds
- **Height of the bottom of the Ferris wheel above ground:** 2 meters

### Part a) Finding the Height Equation

**1. Determine the Center Height (D):**
Since the bottom of the Ferris wheel is 2 meters above the ground and the radius is 20 meters, the center of the Ferris wheel is:
$$ D = 2\, \text{meters} + 20\, \text{meters} = 22\, \text{meters} $$

**2. Amplitude (A):**
The amplitude represents the maximum deviation from the center height, which is equal to the radius:
$$ A = 20\, \text{meters} $$

**3. Angular Frequency (B):**
The angular frequency relates to how quickly the Ferris wheel makes a full rotation:
$$ B = \frac{2\pi}{T} = \frac{2\pi}{180} = \frac{\pi}{90}\, \text{radians per second} $$

**4. Phase Shift (C):**
Since passengers board at the **lowest point** at $$ t = 0 $$, the cosine function (which typically starts at a maximum) needs to be shifted to start at a minimum. This can be achieved by introducing a phase shift of $$ \pi $$:
$$ C = \pi $$

**5. Constructing the Equation:**
Combining these components, the height $$ h(t) $$ as a function of time $$ t $$ is:
$$ h(t) = D - A \cdot \cos(Bt) $$
$$ h(t) = 22\, \text{m} - 20\, \text{m} \cdot \cos\left(\frac{\pi t}{90}\right) $$

**Final Equation:**
$$ h(t) = 22 - 20\,\cos\left(\frac{\pi t}{90}\right) $$
*where $$ h(t) $$ is the height in meters and $$ t $$ is the time in seconds.*

### Answer to Part a):

The height $$ h(t) $$ (in meters) of a passenger at time $$ t $$ (in seconds) is
$$ h(t) = 22 - 20\,\cos\left(\frac{\pi t}{90}\right) $$.
The height of a passenger at time $$ t $$ seconds is given by: $$ h(t) = 22 - 20\,\cos\left(\frac{\pi t}{90}\right) $$

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