Question

Write a sine function where the amplitude is 12 and the revolutions are 70 per minute

Answer 1

To write a sine function with a specific amplitude and frequency, we can start with the general form of the sine function:

\[
y(t) = A \sin(Bt + C) + D
\]

Where:
- \(A\) is the amplitude,
- \(B\) relates to the frequency,
- \(C\) is a phase shift (which we will set to 0 unless specified otherwise),
- \(D\) is a vertical shift (which we will also set to 0 unless specified otherwise).

### Given:
- The amplitude \(A = 12\).
- Revolutions \(= 70\) per minute.

### Frequency:
To relate revolutions per minute to the sine function, we first convert revolutions per minute into radians per second. Since one complete revolution corresponds to \(2\pi\) radians, we convert the frequency as follows:

\[
\text{Frequency (in radians per second)} = 70 \text{ revolutions/minute} \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}}
\]

Calculating that gives:

\[
\text{Frequency} = 70 \times \frac{2\pi}{60} = \frac{140\pi}{60} = \frac{7\pi}{3} \text{ radians/second}
\]

### Setting B:
The value of \(B\) in the sine function is related to the frequency by the formula:

\[
B = \text{Frequency}
\]

Thus, we have:

\[
B = \frac{7\pi}{3}
\]

Now substituting the values into the sine function equation:
- Amplitude \(A = 12\)
- \(B = \frac{7\pi}{3}\)
- Phase shift \(C = 0\)
- Vertical shift \(D = 0\)

### Final Function:
The sine function can now be written as:

\[
y(t) = 12 \sin\left(\frac{7\pi}{3} t\right)
\]

This function describes a sine wave with an amplitude of 12 that completes 70 revolutions per minute.