Answer
$s(t)=-3\cos 12t$
Work Step by Step
The general equation regarding the oscillation of a weight attached to a spring is $s(t)=a\cos wt$.
Since the weight is first pulled down and then released, the equation becomes $s(t)=-a\cos wt$. Also, as the weight is pulled down 3 inches, the amplitude of the motion is 3. The equation therefore becomes $s(t)=-3\cos wt$.
Now we need to find $w$. Since the frequency is $\frac{6}{\pi}$, the period is $\frac{\pi}{6}$ since the two are reciprocals of each other. Now, we can use the following formula to find $w$:
Period$=\frac{2\pi}{w}$
$\pi/6=\frac{2\pi}{w}$
$w=\frac{2\pi}{\pi/6}$
$w=12$
Therefore, the model that gives the position of the weight at time $t$ is $s(t)=-3\cos 12t$.
