Question

The given figure represents a hemisphere surmounted by a conical block of wood. The diameter of their bases is 6 cm each and the slant height of the cone is 5 cm. Calculate:
(i) the height of the cone.
(ii) the volume of the solid. [4 MARKS]

Answer 1

Each Part: 2 Marks

Given: Slant height, l = 5 cm
and diameter = 6 cm
$\therefore$ r = 3 cm

i) Let the height of cone be 'h'
Then, $l^2=h^2+r^2$
$\Rightarrow 5^2=h^2+3^2$
$\Rightarrow h=\sqrt{25-9}cm=4cm$

ii) Volume of solid = Volume of cone + Volume of hemisphere
$=\frac{1}{3}\pi r^{2}h+\frac{2}{3}\pi r^3=\frac{1}{3}\pi r^2(h+2r)$
$=\frac{1}{3}\times \frac{22}{7}\times 9(4+2\times 3)=94.3c{m}^3$
$\therefore$ Volume of the metallic sphere
$=\frac{4}{3}\pi R^3=\frac{4}{3}\pi (7{)}^{3}c{m}^3$
Radius of the small metallic sphere
$=\frac{3.5}{2}=\frac{7}{4}cm$
Volume of a small metallic sphere
$=\frac{4}{3}\pi r^3=\frac{4}{3}\pi {\left(\frac{7}{4}\right)}^{3}c{m}^3$