One-third of a bamboo pole is buried in mud, one-sixth of it is in water and the part above the water is greater than or equal to 3 metres. Find the length of the shortest pole.

Let the length of the shortest pole = x metre

Length of pole which is burried in mud = $\dfrac{x}{3}$

Length of pole which is in the water = $\dfrac{x}{6}$

Given,

$x-\Big[\dfrac{x}{3} + \dfrac{x}{6}\Big] \ge 3 \\[0.5em] \Rightarrow x - \Big(\dfrac{2x + x}{6}\Big) \ge 3 \\[0.5em] \Rightarrow x - \Big(\dfrac{3x}{6}\Big) \ge 3 \\[0.5em] \Rightarrow x - \dfrac{x}{2} \ge 3 \\[0.5em] \Rightarrow \dfrac{x}{2} \ge 3 \\[0.5em] \Rightarrow x \ge 6$

∴ Length of pole which is shortest = 6 meters.