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प्रश्न
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.
उत्तर
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
\[= 6 a^2 - \frac{1}{3} \pi r^2 h\]
\[ = 6 \times \left( 14 \right)^2 - \left[ 154\left( 1 + \sqrt{5} \right) \right]\]
\[ = 1022 + 154\sqrt{5} {cm}^2\]
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