Answer :
To solve the problem of modeling the height of the end of a windmill blade as a sine function, we need to follow certain steps based on the data provided.
1. Understand the Components of the Sine Model:
The general sine model given is [tex]\( y = a \sin(b t) + k \)[/tex], where:
- [tex]\( a \)[/tex] is the amplitude of the sine wave.
- [tex]\( b \)[/tex] is related to the period of the sine wave.
- [tex]\( k \)[/tex] is the vertical shift of the sine wave.
2. Identify the Amplitude [tex]\( a \)[/tex]:
The amplitude [tex]\( a \)[/tex] is the maximum deviation of the wave from its central position, which in this case is equal to the length of the windmill blades.
Therefore, [tex]\( a = 15 \)[/tex] feet.
3. Determine the Vertical Shift [tex]\( k \)[/tex]:
The vertical shift [tex]\( k \)[/tex] is the height of the axis from the ground, which means it shifts the sine wave vertically.
Here, [tex]\( k = 40 \)[/tex] feet.
4. Calculate the Period of the Sine Wave:
Since the windmill completes 3 rotations every minute, it makes 1 rotation in [tex]\( \frac{60}{3} = 20 \)[/tex] seconds.
Hence, the period of one complete rotation, [tex]\( T \)[/tex], is 20 seconds.
5. Calculate [tex]\( b \)[/tex] from the Period:
The relationship between [tex]\( b \)[/tex] and the period is given by the formula [tex]\( T = \frac{2\pi}{b} \)[/tex].
Solving for [tex]\( b \)[/tex], we get [tex]\( b = \frac{2\pi}{T} \)[/tex].
Substituting the period, [tex]\( b = \frac{2\pi}{20} = 0.3141592653589793 \)[/tex].
By gathering all this information, the completed sine model for the height [tex]\( y \)[/tex] of the windmill blade is:
[tex]\[ y = 15 \sin(0.3141592653589793 \cdot t) + 40 \][/tex]
In this model:
- [tex]\( a = 15 \)[/tex] represents the blade length.
- The vertical shift [tex]\( k = 40 \)[/tex] is the height of the axis above the ground.
- The period is 20 seconds, with [tex]\( b = 0.3141592653589793 \)[/tex].
This function models the height of the tip of a windmill blade over time as it rotates.
1. Understand the Components of the Sine Model:
The general sine model given is [tex]\( y = a \sin(b t) + k \)[/tex], where:
- [tex]\( a \)[/tex] is the amplitude of the sine wave.
- [tex]\( b \)[/tex] is related to the period of the sine wave.
- [tex]\( k \)[/tex] is the vertical shift of the sine wave.
2. Identify the Amplitude [tex]\( a \)[/tex]:
The amplitude [tex]\( a \)[/tex] is the maximum deviation of the wave from its central position, which in this case is equal to the length of the windmill blades.
Therefore, [tex]\( a = 15 \)[/tex] feet.
3. Determine the Vertical Shift [tex]\( k \)[/tex]:
The vertical shift [tex]\( k \)[/tex] is the height of the axis from the ground, which means it shifts the sine wave vertically.
Here, [tex]\( k = 40 \)[/tex] feet.
4. Calculate the Period of the Sine Wave:
Since the windmill completes 3 rotations every minute, it makes 1 rotation in [tex]\( \frac{60}{3} = 20 \)[/tex] seconds.
Hence, the period of one complete rotation, [tex]\( T \)[/tex], is 20 seconds.
5. Calculate [tex]\( b \)[/tex] from the Period:
The relationship between [tex]\( b \)[/tex] and the period is given by the formula [tex]\( T = \frac{2\pi}{b} \)[/tex].
Solving for [tex]\( b \)[/tex], we get [tex]\( b = \frac{2\pi}{T} \)[/tex].
Substituting the period, [tex]\( b = \frac{2\pi}{20} = 0.3141592653589793 \)[/tex].
By gathering all this information, the completed sine model for the height [tex]\( y \)[/tex] of the windmill blade is:
[tex]\[ y = 15 \sin(0.3141592653589793 \cdot t) + 40 \][/tex]
In this model:
- [tex]\( a = 15 \)[/tex] represents the blade length.
- The vertical shift [tex]\( k = 40 \)[/tex] is the height of the axis above the ground.
- The period is 20 seconds, with [tex]\( b = 0.3141592653589793 \)[/tex].
This function models the height of the tip of a windmill blade over time as it rotates.
