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, 

\[r = \frac{14}{2} = 7 cm\]

Height of the cone, h = 14 cm

Slant height of the cone, \[l = \sqrt{h^2 + r^2}\]

\[\Rightarrow l = \sqrt{\left( 7 \right)^2 + \left( 14 \right)^2}\]

\[ \Rightarrow l = \sqrt{49 + 196} = \sqrt{245} = 7\sqrt{5} cm\]

Surface area of the cone 

\[= \pi r\left( r + l \right)\]

\[ = \pi\left( 7 \right)\left( 7 + 7\sqrt{5} \right)\]

\[ = 154\left( 1 + \sqrt{5} \right) {cm}^2\]

Surface area of the remaining solid = Surface area of the cube − surface area of the cone

\[= 6 a^2 - \frac{1}{3} \pi r^2 h\]

\[ = 6 \times \left( 14 \right)^2 - \left[ 154\left( 1 + \sqrt{5} \right) \right]\]

\[ = 1022 + 154\sqrt{5} {cm}^2\]