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A hemispherical depression is cut out from one face of the cubical wooden block such that the diameter L of the hemisphere is equal to the edge of th e cube. Determine the surface area of the remaining solid.

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\frac{L^{2}}{4} (π + 25)

\frac{L^{2}}{4} (π + 24)

\frac{L^{3}}{4} (π + 24)

\frac{L^{2}}{4} (π + 26)

answer is B.

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Detailed Solution

Concept- The surface area of required remaining solid can be found out by subtracting the surface area of the hemisphere from cuboid.
The hemisphere is of diameter L which is cut out from the top surface of the cuboid.

∴ Radius of hemisphere (r) = \frac{dia}{2} = \frac{L}{2}

Hemisphere-
Top surface area,

Area 1 = π r^{2} = π {(\frac{L}{2})}^{2} sq . units

Curved surface area,

Area 2 = 2 π r^{2} = 2 π {(\frac{L}{2})}^{2} sq . units

Cuboid-
Surface area,

Area 3 = 6 {(L)}^{2} sq . units

Therefore, required remaining surface area,

A = Area 3 - (Area 1 + Area 2)

A = 6 {(L)}^{2} - π {(\frac{L}{2})}^{2} + 2 π {(\frac{L}{2})}^{2}

A = \frac{L^{2}}{4} (π + 24) sq . units

Hence, the correct option is 2.

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