To write a sine model for the height of the end of one blade as a function of time, we need to analyze the given parameters systematically.
### 1. Understanding the Axis and Blade Height
- The axis of the windmill is located 30 feet above the ground.
- The blades extend 10 feet from the axis.
### 2. Defining Amplitude ([tex]\(a\)[/tex])
- The amplitude of the sine function is the maximum deviation from the center value, which corresponds to the length of the blade.
- Therefore, [tex]\(a = 10\)[/tex] feet.
### 3. Defining Vertical Shift ([tex]\(k\)[/tex])
- The vertical shift represents the height of the axis above the ground because that is the center line about which the blade oscillates.
- Therefore, [tex]\(k = 30\)[/tex] feet.
### 4. Defining Period and Frequency ([tex]\(b\)[/tex])
- The windmill completes 2 rotations every minute.
- Thus, 1 rotation takes [tex]\(30\)[/tex] seconds.
- The period of the sine function, which corresponds to one full rotation, is [tex]\(30\)[/tex] seconds.
Frequency ([tex]\(b\)[/tex]) can be calculated from the period using the formula:
[tex]\[ b = \frac{2\pi}{\text{period}} \][/tex]
[tex]\[ b = \frac{2\pi}{30} \][/tex]
[tex]\[ b = \frac{\pi}{15} \][/tex]
### 5. Writing the Sine Function
Putting it all together, the height [tex]\(y\)[/tex] of the end of one blade as a function of time [tex]\(t\)[/tex] can be modeled by:
[tex]\[ y = a \sin(bt) + k \][/tex]
[tex]\[ y = 10 \sin\left(\frac{\pi}{15} t\right) + 30 \][/tex]
Thus, the correct sine model is:
[tex]\[ y = 10 \sin\left(\frac{\pi}{15} t\right) + 30 \][/tex]
Out of the provided options, the correct one is:
[tex]\[ y = 10 \sin\left(\frac{\pi}{15} t\right) + 30 \][/tex]
