To create a sine model [tex]\( y = a \sin (bt) + k \)[/tex] for the height of the windmill blade as a function of time [tex]\( t \)[/tex] in seconds, we will use the given information:
1. Amplitude ([tex]\(a\)[/tex]):
The amplitude [tex]\(a\)[/tex] represents the length of the blade. Therefore,
[tex]\[ a = 15 \, \text{feet} \][/tex]
2. Vertical Shift ([tex]\(k\)[/tex]):
The vertical shift [tex]\(k\)[/tex] corresponds to the height of the windmill's axis above the ground. Therefore,
[tex]\[ k = 40 \, \text{feet} \][/tex]
3. Period:
The windmill completes 3 rotations every minute. To find the period of one rotation in seconds:
- Since there are 60 seconds in a minute, the period for one complete rotation is:
[tex]\[ \text{Period} = \frac{60 \, \text{seconds}}{3 \, \text{rotations}} = 20 \, \text{seconds} \][/tex]
4. Angular Frequency ([tex]\(b\)[/tex]):
The angular frequency [tex]\(b\)[/tex] is found using the formula [tex]\( b = \frac{2\pi}{\text{Period}} \)[/tex]. With the period of 20 seconds:
[tex]\[
b = \frac{2\pi}{20} = 0.3141592653589793
\][/tex]
Now, let's summarize all the parameters for the sine model [tex]\( y = a \sin (bt) + k \)[/tex]:
- [tex]\(a = 15\)[/tex]
- [tex]\(k = 40\)[/tex]
- The period of the windmill blades is 20 seconds.
- [tex]\(b = 0.3141592653589793\)[/tex]
Putting it all together, the sine model for the height of the windmill blade as a function of time [tex]\( t \)[/tex] in seconds is:
[tex]\[ y = 15 \sin (0.3141592653589793 \, t) + 40 \][/tex]
