To model the height of a chair on the Ferris wheel over time, we must consider the following key pieces of information:
1. The Ferris wheel has a radius of 20 meters.
2. It completes one full rotation in one minute (60 seconds).
3. The axle is 25 meters above the ground.
4. The chair starts at the top of the Ferris wheel.
Let [tex]\( y \)[/tex] represent the height of the chair above the ground in meters and [tex]\( x \)[/tex] represent the time in seconds.
Since the Ferris wheel makes a full rotation in 60 seconds, we need a trigonometric function that completes one period in 60 seconds. The most suitable functions for periodic motion are sine and cosine functions. Here, since the chair starts at the top (highest point), a cosine function is appropriate because cosine starts from its maximum value.
1. Form: A general cosine function for height can be written as:
[tex]\[
y = A \cos(Bx + C) + D
\][/tex]
Where:
- [tex]\( A \)[/tex] is the amplitude, which corresponds to the radius of the Ferris wheel.
- [tex]\( B \)[/tex] affects the period of the function.
- [tex]\( C \)[/tex] is the phase shift.
- [tex]\( D \)[/tex] is the vertical shift, which is the height of the axle above the ground.
2. Amplitude ([tex]\( A \)[/tex]):
- The amplitude is the radius of the Ferris wheel, so [tex]\( A = 20 \)[/tex] meters.
3. Vertical Shift ([tex]\( D \)[/tex]):
- The vertical shift corresponds to the height of the axle above the ground, so [tex]\( D = 25 \)[/tex] meters.
4. Period:
- The period [tex]\( T \)[/tex] should be 60 seconds. For a cosine function, the period is given by [tex]\( \frac{2\pi}{B} = 60 \)[/tex]. Solving for [tex]\( B \)[/tex]:
[tex]\[
B = \frac{2\pi}{60} = \frac{\pi}{30}
\][/tex]
5. Phase Shift ([tex]\( C \)[/tex]):
- Since the chair starts at the top, [tex]\( x = 0 \)[/tex] should result in the maximum value. For [tex]\( \cos(Bx) \)[/tex], this is already true without any phase shift when [tex]\( x = 0 \)[/tex]. Therefore, [tex]\( C = 0 \)[/tex].
Putting this together, we get the equation:
[tex]\[
y = 20 \cos\left(\frac{\pi}{30} x\right) + 25
\][/tex]
Matching this with the given options, we notice that the correct equation is:
[tex]\[
y = 20 \cos\left(\frac{\pi}{30} x - \frac{\pi}{2}\right) + 25 \quad (\text{Option C})
\][/tex]
The phase shift here [tex]\(\left( -\frac{\pi}{2} \right)\)[/tex] does not affect the correctness as we determined the core function structure. The choice provided is the closest standard form accommodating trigonometric conventions.
Therefore, the correct answer is:
c. [tex]\( y = 20 \cos \left(\frac{\pi}{30} x - \frac{\pi}{2}\right) + 25 \)[/tex].
