Answer :
We are given that the windmill’s center of rotation is 40 feet above the ground, the blades are 15 feet long, and the windmill turns through 3 complete rotations every minute.
Let’s break down the steps to obtain the sine model
1. Amplitude, [tex]\( a \)[/tex]:
The amplitude is the distance from the center of the motion to the maximum or minimum value. Since the blade is 15 feet long, the amplitude is
[tex]$$ a = 15. $$[/tex]
2. Vertical Shift, [tex]\( k \)[/tex]:
The vertical shift represents the average height of the end of the blade. As the rotation axis is 40 feet above the ground, the vertical shift is
[tex]$$ k = 40. $$[/tex]
3. Determining the Angular Frequency, [tex]\( b \)[/tex]:
The windmill makes 3 rotations per minute. First, convert the period (time for one rotation) to seconds. The period [tex]\( T \)[/tex] is given by:
[tex]$$ T = \frac{60\text{ seconds}}{3} = 20 \text{ seconds}. $$[/tex]
The angular frequency [tex]\( b \)[/tex] is related to the period by
[tex]$$ b = \frac{2\pi}{T} = \frac{2\pi}{20} = \frac{\pi}{10}. $$[/tex]
4. Constructing the Sine Model:
The model for the height of the end of one blade as a function of time [tex]\( t \)[/tex] (in seconds) is:
[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]
Thus, the amplitude is
[tex]$$ a = 15, $$[/tex]
and the vertical shift is
[tex]$$ k = 40. $$[/tex]
The complete sine model is
[tex]$$ y = 15 \sin\left(\frac{\pi}{10} t\right) + 40. $$[/tex]
Let’s break down the steps to obtain the sine model
1. Amplitude, [tex]\( a \)[/tex]:
The amplitude is the distance from the center of the motion to the maximum or minimum value. Since the blade is 15 feet long, the amplitude is
[tex]$$ a = 15. $$[/tex]
2. Vertical Shift, [tex]\( k \)[/tex]:
The vertical shift represents the average height of the end of the blade. As the rotation axis is 40 feet above the ground, the vertical shift is
[tex]$$ k = 40. $$[/tex]
3. Determining the Angular Frequency, [tex]\( b \)[/tex]:
The windmill makes 3 rotations per minute. First, convert the period (time for one rotation) to seconds. The period [tex]\( T \)[/tex] is given by:
[tex]$$ T = \frac{60\text{ seconds}}{3} = 20 \text{ seconds}. $$[/tex]
The angular frequency [tex]\( b \)[/tex] is related to the period by
[tex]$$ b = \frac{2\pi}{T} = \frac{2\pi}{20} = \frac{\pi}{10}. $$[/tex]
4. Constructing the Sine Model:
The model for the height of the end of one blade as a function of time [tex]\( t \)[/tex] (in seconds) is:
[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]
Thus, the amplitude is
[tex]$$ a = 15, $$[/tex]
and the vertical shift is
[tex]$$ k = 40. $$[/tex]
The complete sine model is
[tex]$$ y = 15 \sin\left(\frac{\pi}{10} t\right) + 40. $$[/tex]
