Question

A cone of maximum size is carved out from a cube of edge 14 cm . Find the surface area of the cone and of the remaining solid left out after the cone carved out .

Answer 1

The base of the largest right circular cone will be the circle inscribed in a face of the cube and its height will be equal to an edge of the cube.

Given, Edge of the cube = 14 cm

Radius of base of the cone, $r=\frac{14}{2}=7cm$

Height of the cone, h = 14 cm

Slant height of the cone, $l=\sqrt{h^2+r^2}$
$$\begin{gathered} \Rightarrow l=\sqrt{{\left(7\right)}^2+{\left(14\right)}^2} \\ \Rightarrow l=\sqrt{49+196}=\sqrt{245}=7\sqrt{5}\mathrm{cm} \end{gathered}$$

Surface area of the cone
$$\begin{gathered} =\mathrm{\pi r}\left(\mathrm{r}+\mathrm{l}\right) \\ =\mathrm{\pi }\left(7\right)\left(7+7\sqrt{5}\right) \\ =154\left(1+\sqrt{5}\right){\mathrm{cm}}^2 \end{gathered}$$
Surface area of the remaining solid = Surface area of the cube − surface area of the cone
$$\begin{gathered} =6a^2-\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h} \\ =6\times {\left(14\right)}^2-\left[154\left(1+\sqrt{5}\right)\right] \\ =1022+154\sqrt{5}{\mathrm{cm}}^2 \end{gathered}$$