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

Answer 1

(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.


$$\begin{gathered} \mathrm{As}R=\frac{\mathrm{Diameter}}{2}, \\ R=\frac{2}{2}\text{cm}=1\text{cm} \\ h=16\text{cm} \end{gathered}$$

Therefore,
$$\begin{gathered} 12\times \frac{4}{3}\mathrm{\pi }{r}^3=\mathrm{\pi }{R}^{2}h \\ \Rightarrow 12\times \frac{4}{3}{r}^3=R^{2}h \end{gathered}$$

$$\begin{gathered} \Rightarrow 12\times \frac{4}{3}{\left(\frac{d}{2}\right)}^3={\left(1\right)}^2\times 16 \\ \Rightarrow 16\times \frac{d^3}{8}=16 \end{gathered}$$
^{$$\begin{gathered} \Rightarrow d^3=8 \\ \Rightarrow d=\pm 2 \\ \mathrm{Since}\mathrm{d}\mathrm{cannot}\mathrm{be}\mathrm{negative},\mathrm{thus},d=2 \end{gathered}$$}

Hence, the diameter of each sphere is 2 cm.