Answer :
To write a sine model for the height of the end of a windmill blade as a function of time, let's break the problem down into its components.
1. Amplitude (a): This is the length of the blade. Since the blades are 15 feet long, the amplitude of the sine wave, which reflects the maximum deviation from the central axis of rotation, is 15. Therefore, [tex]\( a = 15 \)[/tex].
2. Vertical Shift (k): This is the height of the axis of rotation from the ground. Since the axis is 40 feet high, the vertical shift is 40. Therefore, [tex]\( k = 40 \)[/tex].
3. Rotational Speed and Period: The windmill completes 3 rotations every minute. To find the period, which is the time it takes to complete one full rotation, we calculate:
- There are 60 seconds in a minute.
- With 3 rotations per minute, the period for one rotation is [tex]\(\frac{60 \text{ seconds}}{3 \text{ rotations}} = 20 \text{ seconds per rotation}\)[/tex].
4. Frequency and Angular Frequency (b): The frequency tells us how many cycles are completed in one second, which is [tex]\(\frac{1 \text{ rotation}}{20 \text{ seconds}}\)[/tex]. To convert this into angular frequency (b), we use the relationship [tex]\( b = \frac{2\pi}{\text{Period}} \)[/tex].
- Given the period is 20 seconds, we find [tex]\( b = \frac{2\pi}{20} = \frac{\pi}{10} \approx 0.314\)[/tex].
The sine model for the height of the end of one blade as a function of time [tex]\( t \)[/tex] in seconds is given by:
[tex]\[ y = a \sin(b t) + k \][/tex]
Substituting the values we found:
[tex]\[ y = 15 \sin\left(\frac{\pi}{10} t\right) + 40 \][/tex]
This sine model accurately represents the vertical motion of the tip of the windmill blade relative to the ground as it rotates.
1. Amplitude (a): This is the length of the blade. Since the blades are 15 feet long, the amplitude of the sine wave, which reflects the maximum deviation from the central axis of rotation, is 15. Therefore, [tex]\( a = 15 \)[/tex].
2. Vertical Shift (k): This is the height of the axis of rotation from the ground. Since the axis is 40 feet high, the vertical shift is 40. Therefore, [tex]\( k = 40 \)[/tex].
3. Rotational Speed and Period: The windmill completes 3 rotations every minute. To find the period, which is the time it takes to complete one full rotation, we calculate:
- There are 60 seconds in a minute.
- With 3 rotations per minute, the period for one rotation is [tex]\(\frac{60 \text{ seconds}}{3 \text{ rotations}} = 20 \text{ seconds per rotation}\)[/tex].
4. Frequency and Angular Frequency (b): The frequency tells us how many cycles are completed in one second, which is [tex]\(\frac{1 \text{ rotation}}{20 \text{ seconds}}\)[/tex]. To convert this into angular frequency (b), we use the relationship [tex]\( b = \frac{2\pi}{\text{Period}} \)[/tex].
- Given the period is 20 seconds, we find [tex]\( b = \frac{2\pi}{20} = \frac{\pi}{10} \approx 0.314\)[/tex].
The sine model for the height of the end of one blade as a function of time [tex]\( t \)[/tex] in seconds is given by:
[tex]\[ y = a \sin(b t) + k \][/tex]
Substituting the values we found:
[tex]\[ y = 15 \sin\left(\frac{\pi}{10} t\right) + 40 \][/tex]
This sine model accurately represents the vertical motion of the tip of the windmill blade relative to the ground as it rotates.
