When the blades are turning at a constant rate, the height $h$, in meters, at the tip of a specified blade above the ground can be modeled by the following function of $t$, where $t$, is the number of seconds after the blade started rotating. $$H(t)= 25 + 1.25 \cos\left(\frac{\pi}{0.375}t\right).$$

How many seconds does it take the blade to complete one revolution? Is it $0.357$, $0.446, 0.714$, or $3.501$. I know that one revolution of the blade is the period of a sine and/or cosine function which is $2\pi$ or $2(3.1416)$. If I divide $2\pi$ by $\frac{\pi}{0.375}$ I get $0.75$ which is close to answer C. I wonder if that is the right answer. Can I get some help here? Thank you.Pic of a wind turbine

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    $\begingroup$ Please read the MathJaX tutorial $\endgroup$ Aug 13 '17 at 3:04
  • $\begingroup$ The pic is cute! $\endgroup$ Aug 13 '17 at 14:55

In short, you are correct. Here's a quick explanation of what each part of the equation physically represents, and how it would affect the graph of a cosine equation. $$H(t)= A + B \cos\left(Ct\right).$$ $A$ will shift the entire height function up or down, and represents the height of the axle about which the turbine blade rotates.

$B$ will stretch the height function in the $y$ direction, and represents the radius of the wind turbine blade.

$C$ will shrink the height function in the $x$ direction, and is proportional to the speed of the blade.

As you mention, if $C = 1$, then the period of $\cos\left(t\right)$ is $2\pi$. If $C=2$, the angle of the blade increases at twice the speed it used to, so the period of rotation is half what it once was. Expanding upon this observation, you may note that the period of rotation $T$ can be calculated as follows: $$T = \frac{2\pi}{C} = \frac{2\pi}{\frac{\pi}{0.375}}=0.75$$

Your answer of $0.75$ is correct, but choice $C$ is not. Whatever book this exercise is from most likely contains a mistake.


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