Time Triangle

After waiting for $t$ minutes, the hands make an angle of $(150 + \tfrac12t - 6t)^\circ = (150 - \tfrac{11}{2}t)^\circ$ with each other. We want the smallest value of $t$ such that $\begin{aligned} \tfrac12 \times 3.5 \times 5.3 \sin\big(150 - \tfrac{11}{2}t\big)^\circ & = \; 9.24 \\ \sin\big(150 - \tfrac{11}{2}t\big)^\circ & = \; 0.9962264151 \end{aligned}$ Thus we want $150 - \tfrac{11}{2}t \; = \; 180 - \sin^{-1}0.9962264151$ and hence $t = 10.00379894$ . Michael waits for $\boxed{10}$ minutes.