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Answer :
To solve the problem of writing a sine model for the height of the end of a windmill blade as a function of time, let's break it down step-by-step:
1. Determine the Amplitude (a):
- The amplitude represents the length of the blades. Since the blades are 15 feet long, the amplitude [tex]\(a\)[/tex] is 15.
2. Identify the Vertical Shift (k):
- The blades of the windmill rotate around an axis that is 40 feet above the ground. This height is the vertical shift, [tex]\(k\)[/tex], so [tex]\(k = 40\)[/tex].
3. Calculate the Period:
- The windmill completes 3 rotations every minute, meaning in one minute (60 seconds), the windmill makes 3 full rotations.
- To find the period (the time it takes for one full rotation), divide 60 seconds by the number of rotations per minute: [tex]\(\text{Period} = \frac{60 \text{ seconds}}{3}\)[/tex], which gives a period of 20 seconds.
4. Determine the Value of [tex]\(b\)[/tex]:
- The sine function's period is related to the coefficient [tex]\(b\)[/tex] by the formula: [tex]\(\text{Period} = \frac{2\pi}{b}\)[/tex].
- Since we have computed the period as 20 seconds, we can find [tex]\(b\)[/tex] using the relation: [tex]\(b = \frac{2\pi}{20}\)[/tex].
- Solving for [tex]\(b\)[/tex] gives approximately [tex]\(b = 0.314\)[/tex].
5. Write the Sine Model:
- The function for the height [tex]\(y\)[/tex] of the end of one blade at time [tex]\(t\)[/tex] in seconds is given by:
[tex]\[
y = a \sin(bt) + k
\][/tex]
- Substituting in the values we found:
[tex]\[
y = 15 \sin(0.314 t) + 40
\][/tex]
This equation models the height of the end of a windmill blade in terms of time [tex]\(t\)[/tex], where [tex]\(t\)[/tex] is measured in seconds.
1. Determine the Amplitude (a):
- The amplitude represents the length of the blades. Since the blades are 15 feet long, the amplitude [tex]\(a\)[/tex] is 15.
2. Identify the Vertical Shift (k):
- The blades of the windmill rotate around an axis that is 40 feet above the ground. This height is the vertical shift, [tex]\(k\)[/tex], so [tex]\(k = 40\)[/tex].
3. Calculate the Period:
- The windmill completes 3 rotations every minute, meaning in one minute (60 seconds), the windmill makes 3 full rotations.
- To find the period (the time it takes for one full rotation), divide 60 seconds by the number of rotations per minute: [tex]\(\text{Period} = \frac{60 \text{ seconds}}{3}\)[/tex], which gives a period of 20 seconds.
4. Determine the Value of [tex]\(b\)[/tex]:
- The sine function's period is related to the coefficient [tex]\(b\)[/tex] by the formula: [tex]\(\text{Period} = \frac{2\pi}{b}\)[/tex].
- Since we have computed the period as 20 seconds, we can find [tex]\(b\)[/tex] using the relation: [tex]\(b = \frac{2\pi}{20}\)[/tex].
- Solving for [tex]\(b\)[/tex] gives approximately [tex]\(b = 0.314\)[/tex].
5. Write the Sine Model:
- The function for the height [tex]\(y\)[/tex] of the end of one blade at time [tex]\(t\)[/tex] in seconds is given by:
[tex]\[
y = a \sin(bt) + k
\][/tex]
- Substituting in the values we found:
[tex]\[
y = 15 \sin(0.314 t) + 40
\][/tex]
This equation models the height of the end of a windmill blade in terms of time [tex]\(t\)[/tex], where [tex]\(t\)[/tex] is measured in seconds.
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Rewritten by : Barada
