Question

(a) A solid right circular copper cone of height 15 cm and radius 6 cm is melted and recast into smaller copper cones of height 3 cm and radius 2 cm. How many such smaller cones can be made?

(b) Two solid spheres of radii 2 cm and 4 cm are melted and recast into a cone of height 8 cm. Find the radius of the cone so formed. [6 MARKS]

Answer 1

Each Part: 2 Marks

(a) Volume of the bigger cone
$=\frac{1}{3}\times \pi \times 6^2\times 15$cm^3 = 180 \pi ~cm^3
Volume of one smaller cone
$=\frac{1}{3}\times \pi \times 2^2\times 3$cm^3 = 4 \pi ~cm^3
\therefore Number of smaller cones
$=\frac{Volumeofbiggercone}{Volumeofasmallercone}$
$=\frac{180\pi c{m}^3}{4\pi c{m}^3}=45$
Hence, the number of smaller cones is 45


(b) Radii of two solid sphere are 2 cm and 4 cm.
Spheres are melted and recast into a cone of height 8 cm
$\Rightarrow$ Vol. of two spheres = Vol. of one cone
$\Rightarrow \frac{4}{3}\pi r_1^3+\frac{4}{3}\pi r_2^3=\frac{1}{3}\pi R^{2}H$
$\Rightarrow \frac{4}{3}\pi [r_1^3+r_2^3]=\frac{1}{3}\pi R^2\times 8$
$\Rightarrow 4[2^3+4^3]=8R^2$
$\Rightarrow 4\times 72=8R^2$
$\Rightarrow R^2=36cm$
$\Rightarrow R=6cm$
Radius of cone = 6 cm