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Sagot :
To solve this problem, we need to construct a sine function that appropriately models the height of the end of one of the windmill blades as it rotates around the axis. The model we are looking for is of the form [tex]\( y = a \sin (b t) + k \)[/tex].
Here are the steps to determine the constants [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(k\)[/tex]:
### 1. Determine the Amplitude ([tex]\(a\)[/tex]):
In trigonometric functions, the amplitude is the maximum distance from the midline (axis of rotation in this case) to the peak of the sine wave. It represents how far the end of the blade moves above and below the axis height.
- Given that the blade length is 10 feet, this length will be the amplitude.
- Therefore, [tex]\( a = 10 \)[/tex].
### 2. Determine the Vertical Shift ([tex]\(k\)[/tex]):
The vertical shift moves the sine wave up or down. It represents the axis height from the ground.
- The axis height of the windmill is 30 feet from the ground.
- Hence, [tex]\( k = 30 \)[/tex].
### 3. Determine the Frequency ([tex]\(b\)[/tex]):
The frequency determines how many cycles (rotations) the sine function completes in a given period. We need to convert rotations per minute to an angular frequency in terms of radians per second.
- The windmill completes 2 rotations per minute. Converting this to rotations per second:
[tex]\[ \frac{2 \text{ rotations}}{1 \text{ minute}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} = \frac{1}{30} \text{ rotations per second} \][/tex]
- Since one complete rotation is [tex]\(2\pi\)[/tex] radians, the angular frequency ([tex]\(b\)[/tex]) is:
[tex]\[ b = 2\pi \times \frac{1}{30} = \frac{2\pi}{30} = \frac{\pi}{15} \][/tex]
### Combining [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(k\)[/tex]:
With these values, we can write the sine model for the height [tex]\(y\)[/tex] as a function of time [tex]\(t\)[/tex]:
[tex]\[ y = 10 \sin \left(\frac{\pi}{15} t\right) + 30 \][/tex]
This equation represents the height of the end of one blade as it rotates counterclockwise, starting from the right (horizontal position).
Therefore, the correct choice from the given options is:
[tex]\[ y = 10 \sin \left(\frac{\pi}{15}t \right) + 30 \][/tex]
Here are the steps to determine the constants [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(k\)[/tex]:
### 1. Determine the Amplitude ([tex]\(a\)[/tex]):
In trigonometric functions, the amplitude is the maximum distance from the midline (axis of rotation in this case) to the peak of the sine wave. It represents how far the end of the blade moves above and below the axis height.
- Given that the blade length is 10 feet, this length will be the amplitude.
- Therefore, [tex]\( a = 10 \)[/tex].
### 2. Determine the Vertical Shift ([tex]\(k\)[/tex]):
The vertical shift moves the sine wave up or down. It represents the axis height from the ground.
- The axis height of the windmill is 30 feet from the ground.
- Hence, [tex]\( k = 30 \)[/tex].
### 3. Determine the Frequency ([tex]\(b\)[/tex]):
The frequency determines how many cycles (rotations) the sine function completes in a given period. We need to convert rotations per minute to an angular frequency in terms of radians per second.
- The windmill completes 2 rotations per minute. Converting this to rotations per second:
[tex]\[ \frac{2 \text{ rotations}}{1 \text{ minute}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} = \frac{1}{30} \text{ rotations per second} \][/tex]
- Since one complete rotation is [tex]\(2\pi\)[/tex] radians, the angular frequency ([tex]\(b\)[/tex]) is:
[tex]\[ b = 2\pi \times \frac{1}{30} = \frac{2\pi}{30} = \frac{\pi}{15} \][/tex]
### Combining [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(k\)[/tex]:
With these values, we can write the sine model for the height [tex]\(y\)[/tex] as a function of time [tex]\(t\)[/tex]:
[tex]\[ y = 10 \sin \left(\frac{\pi}{15} t\right) + 30 \][/tex]
This equation represents the height of the end of one blade as it rotates counterclockwise, starting from the right (horizontal position).
Therefore, the correct choice from the given options is:
[tex]\[ y = 10 \sin \left(\frac{\pi}{15}t \right) + 30 \][/tex]
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