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 so that the solid is completely submerged in water. If the radius of the cylinder is 14 cm, find the rise in the water level.
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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
⇒ π × (14)2 × x = $\frac{2}{3} \times π \times 7^3 + \frac{1}{3} \times π \times 7^2 \times 4$
⇒ 196πx = $\frac{2}{3} \times π \times 343 + \frac{1}{3} \times π \times 196$
⇒ 196x = $\frac{2}{3} \times 343 + \frac{1}{3} \times 196$
⇒ 196x = $\frac{686 + 196}{3}$
⇒ 196x = $\frac{882}{3}$
⇒ x = $\frac{882}{3 \times 196} = \frac{882}{588}$ = 1.5 cm.
Hence, rise in water level = 1.5 cm