Question

A metallic spherical shell of internal and external diameters 4 cm and 8 cm, respectively, is melted and recast in the form of a cone of base diameter 8 cm. The height of the cone is
(a) 12 cm
(b) 14 cm
(c) 15 cm
(d) 8 cm

Answer 1

(b) 14 cm
Let the internal and external radii of the spherical shell be r and R, respectively.
Volume of the spherical shell$=\frac{4}{3}\mathrm{\pi }\left(R^3-r^3\right)$
$$\begin{gathered} =\frac{4}{3}\mathrm{\pi }\left[{\left(4\right)}^3-{\left(2\right)}^3\right]{\text{cm}}^3\left[\text{Since}\mathrm{r}\Rightarrow \frac{4}{2}=2\text{cm},\mathrm{R}=\frac{8}{2}=4\text{cm}\right] \\ =\left(\frac{4}{3}\mathrm{\pi }\times 56\right){\text{cm}}^3 \end{gathered}$$
Radius of the cone = $\frac{8}{2}=4\mathrm{cm}$
Volume of the cone$=\frac{1}{3}\mathrm{\pi }{r}^{2}h$
$=\left(\frac{1}{3}\mathrm{\pi }\times 4\times 4\times h\right){\text{cm}}^3$
Therefore,
$$\begin{gathered} \frac{1}{3}\mathrm{\pi }\times 4\times 4\times h=\frac{4}{3}\mathrm{\pi }\times 56 \\ \Rightarrow \frac{16}{3}\times h=\frac{4}{3}\mathrm{\pi }\times 56 \\ \Rightarrow h=\frac{4\times 56\times 3}{3\times 16} \\ \Rightarrow h=14\text{cm} \end{gathered}$$