Let's go through the problem step by step to form a sine model for the height of the end of one blade as a function of time [tex]\( t \)[/tex].
1. Height of the Axis from the Ground:
The axis of the windmill is 40 feet above the ground. This gives us the vertical shift (k) in our sine model.
[tex]\[
k = 40 \text{ feet}
\][/tex]
2. Length of the Blades:
Each blade of the windmill is 15 feet long. This represents the amplitude (a) in our sine model, which is the maximum distance the end of the blade moves from the axis.
[tex]\[
a = 15 \text{ feet}
\][/tex]
3. Rotations per Minute:
The windmill completes 3 rotations per minute. To form our sine model, we need to find the angular frequency, which requires knowing the period of one rotation. Since there are 60 seconds in a minute, each rotation takes:
[tex]\[
\text{Time period for one rotation} = \frac{60 \text{ seconds}}{3 \text{ rotations}} = 20 \text{ seconds}
\][/tex]
4. Angular Frequency:
The angular frequency ([tex]\(b\)[/tex]) is calculated as:
[tex]\[
b = \frac{2\pi}{\text{Time period}} = \frac{2\pi}{20} = 0.3141592653589793
\][/tex]
Now, we have all the components needed to write the sine model [tex]\( y = a \sin(b t) + k \)[/tex]:
[tex]\[
y = 15 \sin(0.3141592653589793 \cdot t) + 40
\][/tex]
So, filling in the blanks:
- The amplitude, [tex]\(a\)[/tex], is the length of the blades: [tex]\[15 \text{ feet}\][/tex]
- The vertical shift, [tex]\(k\)[/tex], is the height of the axis from the ground: [tex]\[40 \text{ feet}\][/tex]
- [tex]\(a = 15\)[/tex]
- [tex]\(k = 40\)[/tex]
Thus, the completed sine model for the height of the end of one blade as a function of time [tex]\( t \)[/tex] is:
[tex]\[y = 15 \sin(0.3141592653589793 \cdot t) + 40\][/tex]
