1. **State the problem:** We have a cylindrical tank 14.59 m tall which empties completely in 10.20 minutes through a hole at the bottom. We want to find how long it takes for the water level to drop by 6 m from the top. 2. The flow from the hole is governed by Torricelli's law, which implies the draining time $T$ is proportional to the square root of the initial height $h_0$ of the water, i.e., $T \propto \sqrt{h_0}$. 3. Since the tank empties fully from 14.59 m in 10.20 minutes, we have $T_{full} = 10.20$ min for $h_{full} = 14.59$ m. 4. We want the time $t$ to drop from 14.59 m down to $14.59 - 6 = 8.59$ m. 5. The time to reach height $h$ is $T(h) = T_{full} \left(1 - \sqrt{\frac{h}{h_{full}}}\right)$. 6. Substitute $h=8.59$, $h_{full}=14.59$, and $T_{full}=10.20$: $$t = 10.20 \times \left(1 - \sqrt{\frac{8.59}{14.59}}\right)$$ 7. Leaving the answer unevaluated as requested, the exact time in minutes is: $$t = 10.20 \left(1 - \sqrt{\frac{8.59}{14.59}}\right)$$