To model the height [tex]\( h \)[/tex] of the end of one blade of a windmill as a function of time [tex]\( t \)[/tex], we need to consider several factors:
1. Axis Height: The axis of the windmill is 35 feet above the ground, which will serve as the vertical shift in our sine wave model.
2. Blade Length: Each blade is 10 feet long, which will be the amplitude of our sine wave.
3. Rotation Frequency: The blades complete two rotations every minute. Since 1 minute = 60 seconds, this translates to 2/60 rotations per second, or 1/30 rotations per second.
To construct our sine function, we need to determine the form of the equation:
[tex]\[ h = A \sin(Bt + C) + D \][/tex]
where:
- [tex]\( A = 10 \)[/tex] (the amplitude, or length of each blade),
- [tex]\( D = 35 \)[/tex] (the height of the axis of rotation above the ground),
- [tex]\( C = 0 \)[/tex] (the phase shift, since the blade starts to the right at [tex]\( t = 0 \)[/tex]).
Next, we need to determine [tex]\( B \)[/tex], the angular frequency. The relationship between angular frequency in radians per second and rotations per second can be written as:
[tex]\[ B = 2 \pi \times \text{frequency} \][/tex]
Since our frequency is [tex]\( \frac{1}{30} \)[/tex]:
[tex]\[ B = 2 \pi \times \frac{1}{30} = \frac{2\pi}{30} = \frac{\pi}{15} \][/tex]
Thus, our equation becomes:
[tex]\[ h = 10 \sin\left(\frac{\pi}{15} t\right) + 35 \][/tex]
Therefore, the correct equation among the given options is:
[tex]\[ h = 10 \sin\left(\frac{\pi}{15} t\right) + 35 \][/tex]
So, the answer is:
[tex]\[ \boxed{3} \][/tex]
