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 level.
Let the rise in water level be x cm.
Volume of displaced water = Volume of hemisphere + Volume of cone.
⇒ π(14)²x = \frac{2}{3} × π × 7³ + \frac{1}{3} × π × 7² × 4
⇒ 196πx = \frac{2}{3} × π × 343 + \frac{1}{3} × π × 196
Cancelling π:
⇒ 196x = \frac{686+196}{3}
⇒ 196x = \frac{882}{3}
⇒ x = \frac{882}{3×196}
⇒ x = \frac{882}{588}
⇒ x = 1.5
Therefore, the rise in the water level is 1.5 cm.
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