A hemispherical depression is cut of one face of a cubical wooden block such the diameter l of the hemisphere is equal to the edge of the cube. Determine th surface area of the remaining solid.


Answer :


Edge of the cube =\( l \)

Diameter of the hemisphere = \( l \)

\(\Rightarrow \) Radius of the hemisphere \( = \frac{l}{2} \)

\(\Rightarrow \) Area of the remaining solid after cutting out the hemispherical depression.

\( = \ 6 l^2 \ - \ \pi ( \frac{l}{2} )^2 \ + \ 2\pi (\frac{l}{2})^2 \ \)
\( = \ 6l^2 \ + \ \pi (\frac{l}{2})^2 \)

\(= \ 6 l^2 \ + \ \pi \ × \ \frac{l^2}{4} \ \)
\( = \ \frac{l^2}{4} (24 \ + \ \pi) \)

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