Mei Mei takes a trip to an amusement park and rides a Ferris wheel. The graph below shows the height, in feet above the ground, of her car over time, tt, measured in minutes.
0
Time (in minutes)
Height of the car (in feet)
t
y
0
Time (in minutes)
Height of the car (in feet)
(6.75, 340)
(15.75, 340)
(11.25, 10)
(2.25, 10)
Write an equation in terms of yy, height in feet above the ground, and tt, time in minutes, to represent the given context.
Answer
Attempt 1 out of 2
The number of attempts on this specific problem before you are marked incorrect

y Mei Mei takes a trip to an amusement park and rides a Ferris wheel. The graph below shows the height, in feet above the ground, of her car over time, tt, measured in minutes.
0
Time (in minutes)
Height of the car (in feet)
t
y
0
Time (in minutes)
Height of the car (in feet)
(6.75, 340)
(15.75, 340)
(11.25, 10)
(2.25, 10)
Write an equation in terms of yy, height in feet above the ground, and tt, time in minutes, to represent the given context.
Answer
Attempt 1 out of 2
The number of attempts on this specific problem before you are marked incorrect

y 

To model the height of Mei Mei's Ferris wheel ride as a function of time, we recognize that this is a periodic motion, typical of circular movement like a Ferris wheel. The height of the car follows a sinusoidal pattern over time, which suggests that the equation will be of the form:

\[
y(t) = A \cos(B(t - C)) + D
\]

Where:
- \( A \) is the amplitude (half of the total height difference),
- \( B \) is related to the period (how long it takes to complete one full revolution),
- \( C \) is the horizontal shift (phase shift, where the function starts),
- \( D \) is the vertical shift (average height of the car above the ground).

### Step-by-step Solution:

1. **Find the amplitude \( A \)**:
   - The minimum height is \(10\) feet (at \( t = 2.25 \) and \( t = 11.25 \)).
   - The maximum height is \(340\) feet (at \( t = 6.75 \) and \( t = 15.75 \)).
   - The amplitude is half the difference between the maximum and minimum heights:
   \[
   A = \frac{340 - 10}{2} = 165 \text{ feet}.
   \]

2. **Find the vertical shift \( D \)**:
   - The vertical shift is the average of the maximum and minimum heights:
   \[
   D = \frac{340 + 10}{2} = 175 \text{ feet}.
   \]

3. **Determine the period and \( B \)**:
   - The time it takes to complete one full revolution (period) is the time between successive maximums or minimums. The graph shows that the car reaches a maximum height at \( t = 6.75 \) and again at \( t = 15.75 \). This gives a period of:
   \[
   \text{Period} = 15.75 - 6.75 = 9 \text{ minutes}.
   \]
   The relationship between \( B \) and the period is:
   \[
   B = \frac{2\pi}{\text{Period}} = \frac{2\pi}{9}.
   \]

4. **Find the phase shift \( C \)**:
   - The cosine function starts at its maximum, and since the maximum height is at \( t = 6.75 \), the horizontal shift \( C \) is \( 6.75 \).

### Final Equation:
Substituting these values into the general form of the cosine function:

\[
y(t) = 165 \cos\left(\frac{2\pi}{9}(t - 6.75)\right) + 175
\]

This equation represents the height of Mei Mei's Ferris wheel car over time.

Would you like more details or have any questions?

---

### Relative Questions:
1. How would the equation change if the Ferris wheel were smaller?
2. What if the Ferris wheel rotated twice as fast—how would the period change?
3. How can we model this using a sine function instead of a cosine function?
4. What is the significance of the vertical shift in the context of this problem?
5. How can we find the height of the car at any specific time \( t \)?

### Tip:
Always check for the period and phase shift carefully when dealing with periodic motion problems.

Mei Mei takes a trip to an amusement park and rides a Ferris wheel. The graph below shows the height, in feet above the ground, of her car over time, t, measured in minutes. Write an equation in terms of y, height in feet above the ground, and t, time in minutes, to represent the given context.

Explore how to model the height of a Ferris wheel car over time using trigonometric functions. This problem involves key concepts such as amplitude, period, and phase shift to create a cosine equation representing periodic motion. Suitable for students in Grades 10-12.

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