Answer :
To create a sine model for the height of the end of one blade of the windmill as a function of time [tex]\( t \)[/tex] (in seconds), let's break down the problem and find the components of the sine equation: [tex]\( y = a \sin(b t) + k \)[/tex].
1. Amplitude ([tex]\( a \)[/tex]):
- The amplitude is the maximum vertical distance the end of the blade moves from its central position.
- This distance is equal to the length of the blade, which is 15 feet.
- Therefore, [tex]\( a = 15 \)[/tex].
2. Angular Frequency ([tex]\( b \)[/tex]):
- The blades complete 3 rotations every minute. To find the frequency in terms of seconds, we first convert the rotations per minute to rotations per second:
- [tex]\( \text{Rotations per second} = \frac{3 \text{ rotations/minute}}{60 \text{ seconds/minute}} = 0.05 \text{ rotations/second} \)[/tex].
- The angular frequency [tex]\( b \)[/tex], which is in radians per second, is given by the formula [tex]\( b = 2\pi \times \text{frequency} \)[/tex].
- Therefore, [tex]\( b = 2\pi \times 0.05 = 0.3141592653589793 \)[/tex].
3. Vertical Shift ([tex]\( k \)[/tex]):
- The vertical shift corresponds to the height of the axis above the ground, which in this case is 40 feet.
- Thus, [tex]\( k = 40 \)[/tex].
Combining these components, the sine model for the height [tex]\( y \)[/tex] of the end of one blade as a function of time [tex]\( t \)[/tex] in seconds is:
[tex]\[ y = 15 \sin(0.3141592653589793 \times t) + 40 \][/tex]
This equation provides the height from the ground of the point at the tip of the blade over time, assuming the blade is pointing to the right when [tex]\( t=0 \)[/tex] and rotates counterclockwise.
1. Amplitude ([tex]\( a \)[/tex]):
- The amplitude is the maximum vertical distance the end of the blade moves from its central position.
- This distance is equal to the length of the blade, which is 15 feet.
- Therefore, [tex]\( a = 15 \)[/tex].
2. Angular Frequency ([tex]\( b \)[/tex]):
- The blades complete 3 rotations every minute. To find the frequency in terms of seconds, we first convert the rotations per minute to rotations per second:
- [tex]\( \text{Rotations per second} = \frac{3 \text{ rotations/minute}}{60 \text{ seconds/minute}} = 0.05 \text{ rotations/second} \)[/tex].
- The angular frequency [tex]\( b \)[/tex], which is in radians per second, is given by the formula [tex]\( b = 2\pi \times \text{frequency} \)[/tex].
- Therefore, [tex]\( b = 2\pi \times 0.05 = 0.3141592653589793 \)[/tex].
3. Vertical Shift ([tex]\( k \)[/tex]):
- The vertical shift corresponds to the height of the axis above the ground, which in this case is 40 feet.
- Thus, [tex]\( k = 40 \)[/tex].
Combining these components, the sine model for the height [tex]\( y \)[/tex] of the end of one blade as a function of time [tex]\( t \)[/tex] in seconds is:
[tex]\[ y = 15 \sin(0.3141592653589793 \times t) + 40 \][/tex]
This equation provides the height from the ground of the point at the tip of the blade over time, assuming the blade is pointing to the right when [tex]\( t=0 \)[/tex] and rotates counterclockwise.
