Question

A cone of radius 7 cm and height 28 cm is recast into following shapes. Pair the radius in each case.

• A. 7 cm
• B. 3.5 cm
• C. 213 cm

Answer 1

When a solid is melted and moulded to form another solid shape, the volume of the solid formed will be equal to the volume of the solid melted.

Given: Radius of cone, r = 7 cm
Height of the cone, h = 28 cm

$\Rightarrow$ Volume of the cone
= Number of shapes$\times$ Volume of the solid shape formed

1 Sphere:

Let the radius of sphere be R.
Volume of the cone = Number of spheres $\times$ Volume of sphere
$\Rightarrow$ $\frac{1}{3}\pi r^{2}h=1\times \frac{4}{3}\pi R^3$
$\Rightarrow$ $R^3=\frac{r^{2}h}{4}$
$\Rightarrow$ $R^3=\frac{7\times 7\times 28}{4}$
$\Rightarrow$ $R=7cm$
$\therefore$ Radius of the sphere thus formed will be 7 cm.

16 Hemispheres:

Let the radius of hemisphere be R.
Volume of the cone = Number of hemispheres $\times$ Volume of hemisphere
$\Rightarrow$ $\frac{1}{3}\pi r^{2}h=16\times \frac{2}{3}\pi R^3$
$\Rightarrow$ $R^3=\frac{r^{2}h}{2\times 16}$

$\Rightarrow$ $R^3=\frac{7\times 7\times 28}{2\times 16}$

$\Rightarrow$ $R=\frac{7}{2}cm=3.5cm$
$\therefore$ Radius of each hemisphere thus formed will be $3.5cm$.

27 Spheres:

Let the radius of sphere be R.
Volume of the cone = Number of spheres $\times$ Volume of sphere
$\Rightarrow$ $\frac{1}{3}\pi r^{2}h=27\times \frac{4}{3}\pi R^3$
$\Rightarrow$ $R^3=\frac{r^{2}h}{4\times 27}$

$\Rightarrow$ $R^3=\frac{7\times 7\times 28}{4\times 27}$

$\Rightarrow$ $R=\frac{7}{3}cm=2\frac{1}{3}cm$
$\therefore$ Radius of each sphere thus formed will be $2\frac{1}{3}cm$.