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Math Help - Sinusoidal function help

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    Sinusoidal function help

    A small windmill has its center 10 meters above the ground and blades 4 meters in length. In a steady wind, a point P at the tip of one blade makes a rotation in 24 seconds.

    If the rotation begins at the highest possible point, determine a function that gives the height of point P above the ground at a time T
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  2. #2
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    Quote Originally Posted by imakeitrain View Post
    A small windmill has its center 10 meters above the ground and blades 4 meters in length. In a steady wind, a point P at the tip of one blade makes a rotation in 24 seconds.

    If the rotation begins at the highest possible point, determine a function that gives the height of point P above the ground at a time T
    1. Draw a sketch.

    2. According to my sketch the point P has the coordinates P(x, 10+y).

    3. Since the argument of the sine function is an angle measured against(?) the horizontal axis the argument here must be \frac \pi2 + \mu when you calculate the x-coordinate.

    4. According to the text of the question a blade of the mill makes a full turn in 24 s. If T is measured in seconds the angle \mu is calculated by:

    \mu=\frac{2 \pi}{24} \cdot T =\frac{ \pi}{12} \cdot T

    5. Since the argument of the cosine function is an angle measured against(?) the horizontal axis the argument here must be -\frac \pi2 + \mu when you calculate the y-coordinate.

    6. Therefore you have:

    f(T)=\left\{\begin{array}{l}x=4 \cdot \sin\left(\frac \pi2 + \frac{ \pi}{12} \cdot T\right) \\<br />
h=10+y=10+4 \cdot \cos\left(-\frac \pi2 + \frac{ \pi}{12} \cdot T\right)\end{array}\right.

    7. Since you are only interested in the height above ground you'll get:

    h(T)=10+4 \cdot \cos\left(-\frac \pi2 + \frac{ \pi}{12} \cdot T\right)
    Attached Thumbnails Attached Thumbnails Sinusoidal function help-windmuehle.png  
    Last edited by earboth; June 20th 2010 at 11:18 PM.
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    Nicely done, earboth!
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