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Sagot :
Certainly! Let's solve the problem of finding the height function \( f(t) \) for a Ferris wheel.
Given:
- The Ferris wheel's diameter is 40 meters.
- One full rotation takes 8 minutes.
- At \( t=0 \), you are at the 9 o'clock position and descending.
We are to find \( f(t) \), the height of a person above the ground at time \( t \).
### Step-by-Step Solution
1. Determine the radius of the Ferris wheel:
The radius \( r \) is half of the diameter.
[tex]\[ r = \frac{40}{2} = 20 \text{ meters} \][/tex]
2. Calculate the angular speed (\( \omega \)):
The Ferris wheel completes one full rotation ( \( 2\pi \) radians) in 8 minutes.
[tex]\[ \omega = \frac{2\pi \text{ radians}}{8 \text{ minutes}} = \frac{\pi}{4} \text{ radians per minute} \][/tex]
3. Initial position:
At \( t=0 \), the passenger is at the 9 o'clock position, which is mathematically the angle \( \frac{3\pi}{2} \) radians.
This is because:
- 12 o'clock corresponds to angle \( \pi \) radians.
- 9 o'clock is \( \frac{3\pi}{2} \) radians (270 degrees).
4. Angle as a function of time:
Since the wheel rotates counterclockwise and the passenger is descending initially, the angular position at time \( t \) can be expressed as:
[tex]\[ \theta(t) = \frac{3\pi}{2} - \frac{\pi}{4}t \][/tex]
5. Height calculation:
The height above the ground \( h \) in terms of the angle \( \theta \) is determined by the following cosine function, where \( r \) is the radius:
[tex]\[ h = r(1 - \cos(\theta)) \][/tex]
Plug in the expression for \( \theta(t) \):
[tex]\[ h = 20 (1 - \cos\left(\frac{3\pi}{2} - \frac{\pi}{4}t\right)) \][/tex]
### Final Formula
Thus, the height function \( f(t) \) for the Ferris wheel can be written as:
[tex]\[ f(t) = 20 \left(1 - \cos\left(\frac{3\pi}{2} - \frac{\pi}{4}t\right)\right) \][/tex]
This represents your height in meters above the ground at time [tex]\( t \)[/tex] minutes as you ride the Ferris wheel.
Given:
- The Ferris wheel's diameter is 40 meters.
- One full rotation takes 8 minutes.
- At \( t=0 \), you are at the 9 o'clock position and descending.
We are to find \( f(t) \), the height of a person above the ground at time \( t \).
### Step-by-Step Solution
1. Determine the radius of the Ferris wheel:
The radius \( r \) is half of the diameter.
[tex]\[ r = \frac{40}{2} = 20 \text{ meters} \][/tex]
2. Calculate the angular speed (\( \omega \)):
The Ferris wheel completes one full rotation ( \( 2\pi \) radians) in 8 minutes.
[tex]\[ \omega = \frac{2\pi \text{ radians}}{8 \text{ minutes}} = \frac{\pi}{4} \text{ radians per minute} \][/tex]
3. Initial position:
At \( t=0 \), the passenger is at the 9 o'clock position, which is mathematically the angle \( \frac{3\pi}{2} \) radians.
This is because:
- 12 o'clock corresponds to angle \( \pi \) radians.
- 9 o'clock is \( \frac{3\pi}{2} \) radians (270 degrees).
4. Angle as a function of time:
Since the wheel rotates counterclockwise and the passenger is descending initially, the angular position at time \( t \) can be expressed as:
[tex]\[ \theta(t) = \frac{3\pi}{2} - \frac{\pi}{4}t \][/tex]
5. Height calculation:
The height above the ground \( h \) in terms of the angle \( \theta \) is determined by the following cosine function, where \( r \) is the radius:
[tex]\[ h = r(1 - \cos(\theta)) \][/tex]
Plug in the expression for \( \theta(t) \):
[tex]\[ h = 20 (1 - \cos\left(\frac{3\pi}{2} - \frac{\pi}{4}t\right)) \][/tex]
### Final Formula
Thus, the height function \( f(t) \) for the Ferris wheel can be written as:
[tex]\[ f(t) = 20 \left(1 - \cos\left(\frac{3\pi}{2} - \frac{\pi}{4}t\right)\right) \][/tex]
This represents your height in meters above the ground at time [tex]\( t \)[/tex] minutes as you ride the Ferris wheel.
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