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(a) 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.
OR
(b) A cubical block is surmounted by a hemisphere of radius $3.5\text{ cm}$. What is the smallest possible length of the edge of the cube so that the hemisphere can totally lie on the cube ? Find the total surface area of the solid so formed.
OR
(b) A cubical block is surmounted by a hemisphere of radius $3.5\text{ cm}$. What is the smallest possible length of the edge of the cube so that the hemisphere can totally lie on the cube ? Find the total surface area of the solid so formed.
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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 $523.81\text{ cu. cm}$
OR
(b) Edge of cube $= a = 3.5 \times 2 = 7\text{ cm}$
Total surface area of solid
$= 6 a^2 + 2\pi r^2 - \pi r^2$
$= 6 a^2 + \pi r^2$
$= 6 \times 7 \times 7 + \frac{22}{7} \times 3.5 \times 3.5$
$= \frac{665}{2}\text{ sq. cm}$ or $332.5\text{ sq. cm}$
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 $523.81\text{ cu. cm}$
OR
(b) Edge of cube $= a = 3.5 \times 2 = 7\text{ cm}$
Total surface area of solid
$= 6 a^2 + 2\pi r^2 - \pi r^2$
$= 6 a^2 + \pi r^2$
$= 6 \times 7 \times 7 + \frac{22}{7} \times 3.5 \times 3.5$
$= \frac{665}{2}\text{ sq. cm}$ or $332.5\text{ sq. cm}$