A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of π.
π cm³
Explanation
This question is from Exercise 12.2, Q1. The solid consists of two parts: a hemisphere and a cone mounted on it. From Example 7 in the context, we see that the volume of such a toy/solid is calculated as: Volume = (2/3)πr³ + (1/3)πr²h, where the first term is the volume of the hemisphere and the second term is the volume of the cone. Both shapes have the same radius r = 1 cm, and the height of the cone h = 1 cm (equal to its radius as given).
Solution Steps
Given: Radius of hemisphere (r) = 1 cm, Radius of cone (r) = 1 cm, Height of cone (h) = 1 cm
Volume of hemisphere = (2/3)πr³ = (2/3)π(1)³ = (2/3)π cm³
Volume of cone = (1/3)πr²h = (1/3)π(1)²(1) = (1/3)π cm³
Total volume of solid = Volume of hemisphere + Volume of cone = (2/3)π + (1/3)π = π cm³