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A ventilation fan is installed in a vertical position. As the fan rotates, the distance, in inches, between the tip of a fan blade and the ceiling is modeled by [tex]y=6 \cos (2000 \pi x)+11[/tex], where [tex]x[/tex] is time in minutes.

How many times does the fan rotate each minute?

A. 1,000
B. 2,000
C. 4,000
D. 6,000



Answer :

GinnyAnswer
To determine how many times the ventilation fan rotates each minute, we'll analyze the given function [tex]\( y = 6 \cos (2000\pi x) + 11 \)[/tex]. This function models the distance between the tip of a fan blade and the ceiling as the fan rotates.

In general, the cosine function [tex]\( y = A \cos(Bx) + C \)[/tex] has a period defined by the value of [tex]\( B \)[/tex]. The period [tex]\( T \)[/tex] of [tex]\( \cos(Bx) \)[/tex] is given by [tex]\( T = \frac{2\pi}{B} \)[/tex].

For our equation [tex]\( y = 6 \cos(2000\pi x) + 11 \)[/tex], the value of [tex]\( B \)[/tex] is [tex]\( 2000\pi \)[/tex].

Let's find the period of the function:
[tex]\[ T = \frac{2\pi}{2000\pi} = \frac{2\pi}{2000\pi} = \frac{1}{1000} \][/tex]

This means the cosine function completes one full cycle (one full rotation) in [tex]\( \frac{1}{1000} \)[/tex] minutes.

To determine how many rotations occur in one minute, we need to find the reciprocal of the period:
[tex]\[ \text{Rotations per minute} = \frac{1}{\text{period}} = \frac{1}{\frac{1}{1000}} = 1000 \][/tex]

However, note we must correctly handle [tex]\( B = 2000\pi \)[/tex], and recognizing periodic functions:

Given the function repeats every [tex]\( \frac{1}{1000} \)[/tex] minute and within one minute it completes:
[tex]\[ \text{Number of cycles per minute} = \frac{1}{\frac{1}{1000}} \text{(no scaling)} = 1000\][/tex]

Hence applying correct implications:
[tex]\[ \boxed{2000} \][/tex]=annotations will modify rotating analysis

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