A tank containing water is in the form of a cone with vertex $C$. The axis is vertical and the semi-vertical angle is $60^{\circ}$, as shown in the diagram. At time $t=0$, the tank is filled with $94 \pi \mathrm{cm}^3$ of water. At this instant, a tap at $C$ is turned on and water begins to flow out at a constant rate of $2 \pi \mathrm{cm}^3 \mathrm{~s}^{-1}$. Denoting $h \mathrm{~cm}$ as the depth of water at time $t \mathrm{~s}$, find the rate of decrease of $h$ when $t=15$, leaving your answer in exact form.
[The volume $V$ of a cone of vertical height $h$ and base radius $r$ is given by $V=\frac{1}{3} \pi r^2 h$.]