Bot GPT 4-o mini To write a sine or cosine function with the given parameters, we can use the general form of a sine or cosine function:
\[ y = A \cdot \sin(B(x - C)) + D \] or \[ y = A \cdot \cos(B(x - C)) + D \]
where:
- \(A\) is the amplitude,
- \(B\) is related to the period of the function,
- \(C\) is the phase shift (not specified here, so we can assume zero),
- \(D\) is the midline.
Given:
- Amplitude \(A = 12\)
- Midline \(D = 12\)
- Revolutions \(= 70\) per minute
First, let's find the period of the function. The period \(P\) can be calculated as follows:
\[ P = \frac{60}{\text{revolutions per minute}} = \frac{60}{70} \text{ minutes} = \frac{6}{7} \text{ minutes} \]
Now we relate the period to \(B\):
\[ P = \frac{2\pi}{B} \] Thus,
\[ B = \frac{2\pi}{P} = \frac{2\pi}{\frac{6}{7}} = \frac{2\pi \cdot 7}{6} = \frac{7\pi}{3} \]
Assuming no phase shift (\(C = 0\)), we can write the cosine function (for simplicity, since cosine starts at its maximum):
\[ y = 12 \cdot \cos\left(\frac{7\pi}{3}x\right) + 12 \]
So, one possible function is:
\[ y = 12 \cdot \cos\left(\frac{7\pi}{3}x\right) + 12 \]
Alternatively, if you wish to use the sine function, it could be:
\[ y = 12 \cdot \sin\left(\frac{7\pi}{3}x + \frac{\pi}{2}\right) + 12 \]
Both of these functions have the specified amplitude, midline, and represent the system described.
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