Answer
\begin{align}
t \approx 3.73s
\end{align}
Work Step by Step
$\text {The velocity of the mouse is given by}$
\begin{align}
v(t) = 3 \cos {\frac{\pi t}{12}} - 0.5t; 0\leq t \leq 8
\end{align}
$\text {The position of the mouse can be found by}$
\begin{align}
s(t) = \int v(t) \ dt = \frac{36}{\pi} \sin {\frac{\pi t}{12}} -0.25t^2 + C
\end{align}
$\text {Assume at t = 0, the mouse is at s = 0. Therefore:}$
\begin{align}
&C = 0 \\
s(t) = \frac{36}{\pi}& \sin {\frac{\pi t}{12}} -0.25t^2
\end{align}
$\text {We should graph this function to estimate the time. Check attached file.}$
$\text {From the graph:}$
\begin{align}
t \approx 3.73s
\end{align}
