**Question:**

Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter 2 cm and height 16 cm. The diameter of each sphere is

(a) 2 cm

(b) 3 cm

(c) 4 cm

(d) 6 cm

**Solution:**

(a) 2 cm

Let the diameter of each sphere be *d *cm*.*

Let *r* and *R *be the radii of the sphere and the cylinder, respectively,

and *h* be the height of the cylinder.

As $R=\frac{\text { Diameter }}{2}$,

$R=\frac{2}{2} \mathrm{~cm}=1 \mathrm{~cm}$

$h=16 \mathrm{~cm}$

Therefore,

$12 \times \frac{4}{3} \pi r^{3}=\pi R^{2} h$

$\Rightarrow 12 \times \frac{4}{3} r^{3}=R^{2} h$

$\Rightarrow 12 \times \frac{4}{3}\left(\frac{d}{2}\right)^{3}=(1)^{2} \times 16$

$\Rightarrow 16 \times \frac{d^{3}}{8}=16$

$\Rightarrow d^{3}=8$

$\Rightarrow d=\pm 2$

Since $d$ cannot be negative, thus, $d=2$

Hence, the diameter of each sphere is 2 cm.