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The equation [tex]h = 7 \sin \left( \frac{\pi}{21} t \right) + 28[/tex] can be used to model the height, [tex]h[/tex], in feet of the end of one blade of a windmill turning on an axis above the ground as a function of time, [tex]t[/tex], in seconds. How long is the blade? Assume that the blade is pointing to the right, parallel to the ground, at [tex]t = 0[/tex], and that the windmill turns counterclockwise at a constant rate.

A. 7 feet
B. 14 feet
C. 21 feet
D. 28 feet



Answer :

GinnyAnswer
To determine the length of the blade based on the given equation [tex]\( h = 7 \sin \left( \frac{\pi}{21} t \right) + 28 \)[/tex], let's analyze the properties of this periodic function.

1. Understanding the Equation:
- [tex]\( h \)[/tex]: Height of the end of the blade above the ground.
- [tex]\( 7 \)[/tex]: The amplitude of the sine wave, which indicates the maximum deviation from the central value (28 feet).
- [tex]\( \sin \left( \frac{\pi}{21} t \right) \)[/tex]: A sine function that oscillates between −1 and 1.
- [tex]\( 28 \)[/tex]: The vertical shift, which sets the central height above the ground.

2. Finding the Maximum Height:
- The maximum value of [tex]\( \sin \left( \frac{\pi}{21} t \right) \)[/tex] is [tex]\( 1 \)[/tex].
- Substitute [tex]\( \sin \left( \frac{\pi}{21} t \right) = 1 \)[/tex] into the equation:
[tex]\[ h_{\text{max}} = 7 \cdot 1 + 28 = 35 \text{ feet} \][/tex]

3. Finding the Minimum Height:
- The minimum value of [tex]\( \sin \left( \frac{\pi}{21} t \right) \)[/tex] is [tex]\( -1 \)[/tex].
- Substitute [tex]\( \sin \left( \frac{\pi}{21} t \right) = -1 \)[/tex] into the equation:
[tex]\[ h_{\text{min}} = 7 \cdot (-1) + 28 = 21 \text{ feet} \][/tex]

4. Calculating the Blade Length:
- The length of the blade is the difference between the maximum height and the minimum height:
[tex]\[ \text{Blade Length} = h_{\text{max}} - h_{\text{min}} = 35 \text{ feet} - 21 \text{ feet} = 14 \text{ feet} \][/tex]

Therefore, the blade is 14 feet long.

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