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

Given:

A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter $l$ of the hemisphere is equal to the edge of the cube.

To do:

We have to find the surface area of the remaining solid.

Solution:

Edge of the cubical wooden block $a= l$

This implies,

Diameter of the hemisphere curved out of the cube $=l$

Radius of the hemisphere $r=\frac{l}{2}$

Therefore,

Volume of the cube $=a^{3}$

$=(l)^{3}$

$=l^3$ cubic units

Volume of the hemisphere $=\frac{2}{3} \pi r^{3}$

$=\frac{2}{3} \times \pi \times (\frac{l}{2})^3$

$=\frac{1}{12} \pi l^3$ cubic units

Total surface area of the remaining block $=6 a^{2}+2 \pi r^{2}-\pi r^{2}$

$=6 a^{2}+\pi r^{2}$

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

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

$=l^2(\frac{\pi+24}{4})$ square units

The total surface area of the remaining block is $l^2(\frac{\pi+24}{4})$ square units.

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Updated on: 10-Oct-2022

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