Answer
$t=4k+\frac{2}{3}$ and $t=4k+\frac{10}{3}$ seconds.
Work Step by Step
Step 1. Given $d=-6\ cos(\frac{\pi}{2}t)$, for the ball at 3 inches below its rest position, we have $d=-3$ which sets up the equation $-6\ cos(\frac{\pi}{2}t)=-3$ or $ cos(\frac{\pi}{2}t)=\frac{1}{2}$
Step 2. We can find the solutions in $[0,2\pi)$ as $\frac{\pi}{2}t=\frac{\pi}{3}, \frac{5\pi}{3}$
Step 3. We can write all the solutions as $\frac{\pi}{2}t=2k\pi+\frac{\pi}{3}$ and $\frac{\pi}{2}t=2k\pi+\frac{5\pi}{3}$ where $k$ is an integer.
Step 4. Thus, we have all the solutions as $t=4k+\frac{2}{3}$ and $t=4k+\frac{10}{3}$ seconds.
