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Examples 41 to 43 of Mensuration

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(41) Find the volume, curved surface area and total surface area of a hemisphere of diameter 42 cm.

Soln. Volume = $\frac{2}{3}$ r³ = $\frac{2}{3} \times \frac{22}{7} \times 21 \times 21 \times 21$ = 19404 cm³.

Curved surface area = 2πr² = $2 \times \frac{22}{7} \times 21 \times 21$ = 2772 cm².

Total surface area = 3πr² = $3 \times \frac{22}{7} \times 21 \times 21$ = 4158 cm².

(42) How many bullets can be made out of a lead cylinder 14 cm high and 18 cm radius, each bullet being 6 cm in diameter.

Soln.

Number of bullets

= $\frac{\mathrm{Volume\ of\ \ cylinder\ \ in\ c}\mathrm{m}^3}{\mathrm{Volume\ of\ \ 1\ \ bullet\ \ in\ c}\mathrm{m}^3}$

= $\frac{\pi\times18\times18\times14}{\frac{4}{3}\times\pi\times3\times3\times3} = 126.$

(43) Find the number of lead balls of diameter 1 cm each that can be made from a sphere of diameter 8 cm.

Soln.

Number of balls

= $\frac{\mathrm{Volume\ of\ big\ sphere}}{\mathrm{Volume\ of\ \ 1\ \ small\ ball}}$

= $\frac{\frac{4}{3}\pi\times4\times4\times4}{\frac{4}{3}\pi\times0.5\times0.5\times0.5}$

= 512.