Question

A spring bounces up and down according to the model d(t) = 4sin(30t) + 3, were d(t) is the displacement in cm from the rest position and t is time in seconds. What is the equation of the axisÉ

Answer 1

In the given model \( d(t) = 4\sin(30t) + 3 \), the equation describes a sinusoidal motion. The general form of a sinusoidal function can be written as:

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

Where:
- \( A \) is the amplitude,
- \( B \) affects the period,
- \( C \) is the phase shift,
- \( D \) is the vertical shift or midline of the function.

In your function \( d(t) = 4\sin(30t) + 3 \):

- The amplitude \( A = 4 \).
- The coefficient \( 30 \) in front of \( t \) (i.e., \( B = 30 \)), which determines the period of the sine function.
- There is no phase shift (\( C = 0 \)).
- The vertical shift \( D = 3 \).

The equation of the axis (also known as the midline) for this sinusoidal function is given by the vertical shift \( D \).

Thus, the equation of the axis is:

\[
d = 3 \, \text{cm}
\]

This means that the spring oscillates around the position \( 3 \, \text{cm} \) from the rest position.