Question

Given the amplitude is 12, the revolutions are 70 per minute and the midline is 12, write a sine or cosine function

Answer 1

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.