A solid is in the shape of a hemisphere of radius 7 cm, surmounted by a cone of height 4 cm. The solid is immersed completely in a cylindrical container filled with water to a certain height. If the radius of the cylinder is 14 cm, find the rise in the water.

Radius of hemisphere, r = 7 cm

Height of cone, h = 4 cm

Radius of cylinder, R = 14 cm

Let the rise in water level be x cm.

∴ Volume of water that rises by x cm in the cylindrical container = Volume of hemisphere submerged + Volume of cone submerged

$\Rightarrow π\text{R}^2\text{x} = \dfrac{2}{3} π\text{r}^3 + \dfrac{1}{3} π\text{r}^2\text{h} \\[1em] \Rightarrow \text{R}^2\text{x} = \dfrac{1}{3} (2\text{r}^3 + \text{r}^2\text{h}) \\[1em] \Rightarrow 3 \times 14^2\text{x} = 2 \times 7^3 + 7^2 \times 4 \\[1em] \Rightarrow 3 \times 196 \text{x} = 2 \times 343 + 49 \times 4 \\[1em] \Rightarrow 588 \text{x} = 686 + 196 \\[1em] \Rightarrow 588 \text{x} = 882 \\[1em] \Rightarrow \text{x} = \dfrac{882}{588} \\[1em] \Rightarrow \text{x} = 1.5 \text{ cm.}$

Hence, rise in water level is 1.5 cm.