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A toy is in the form of a cone surmounted on a hemisphere. The cone and hemisphere have the same radii. The height of the conical part of the toy is equal to the diameter of its base. If the radius of the conical part is $5\text{ cm}$, find the volume of the toy.
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Solution: (a) Radius $= r = 5\text{ cm}$
Height of cone $= h = 10\text{ cm}$
Volume of toy $=$ volume of hemisphere $+$ volume of cone
$= \frac{2}{3}\pi r^3 + \frac{1}{3}\pi r^2 h$
$= \frac{2}{3} \times \frac{22}{7} \times 5 \times 5 \times 5 + \frac{1}{3} \times \frac{22}{7} \times 5 \times 5 \times 10$
$= \frac{5500}{21} + \frac{5500}{21} = \frac{11000}{21}\text{ cu. cm}$ or $523.81\text{ cu. cm}$
Height of cone $= h = 10\text{ cm}$
Volume of toy $=$ volume of hemisphere $+$ volume of cone
$= \frac{2}{3}\pi r^3 + \frac{1}{3}\pi r^2 h$
$= \frac{2}{3} \times \frac{22}{7} \times 5 \times 5 \times 5 + \frac{1}{3} \times \frac{22}{7} \times 5 \times 5 \times 10$
$= \frac{5500}{21} + \frac{5500}{21} = \frac{11000}{21}\text{ cu. cm}$ or $523.81\text{ cu. cm}$