A conical vessel, whose internal radius is 12 cm and height 50 cm, is full of liquid. The contents are emptied into a cylindrical vessel with internal radius 10 cm. Find the height to which the liquid rises in the cylindrical vessel.

For cylindrical vessel,

Let height be H cm

Radius (R) = 10 cm

For conical vessel,

Height (h) = 50 cm

Radius (r) = 12 cm

Since, contents in conical vessel are emptied into a cylindrical vessel, hence there volume will be same.

∴ Volume of cylinder = Volume of conical vessel

$\Rightarrow π\text{R}^2\text{H} = \dfrac{1}{3}π\text{r}^2\text{h} \\[1em] \Rightarrow \text{R}^2\text{H} = \dfrac{1}{3}\text{r}^2\text{h} \\[1em] \Rightarrow 10^2 \times \text{H} = \dfrac{1}{3} \times 12^2 \times 50 \\[1em] \Rightarrow 100 \times \text{H} = \dfrac{1}{3} \times 144 \times 50 \\[1em] \Rightarrow \text{H} = \dfrac{144 \times 50}{3 \times 100} \\[1em] \Rightarrow \text{H} = \dfrac{7200}{300} \\[1em] \Rightarrow \text{H} = 24 \text{ cm.}$

Hence, height to which the liquid rises in the cylindrical vessel is 24 cm.