A hemispherical depression is cut out from one face of a cubical wooden block of edge $ 21 \mathrm{~cm} $, such that the diameter of the hemisphere is equal to the edge of the cube. Determine the volume and total surface area of the remaining block.

Given:

A hemispherical depression is cut out from one face of a cubical wooden block of edge \( 21 \mathrm{~cm} \), such that the diameter of the hemisphere is equal to the edge of the cube.

To do:

We have to find the volume and total surface area of the remaining block.

Solution:

Edge of the cubical wooden block $a= 21\ cm$

This implies,

Diameter of the hemisphere curved out of the cube $=21 \mathrm{~cm}$

Radius of the hemisphere $r=\frac{21}{2} \mathrm{~cm}$

Therefore,

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

$=(21)^{3}$

$=9261 \mathrm{~cm}^{3}$

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

$=\frac{2}{3} \times \frac{22}{7} \times (\frac{21}{2})^3$

$=\frac{4851}{2}$

$=2425.5 \mathrm{~cm}^{3}$

Volume of the remaining block $=$ Volume of the cubical block $-$ Volume of the hemisphere

$=9261-2425.5$

$=6835.5 \mathrm{~cm}^{3}$

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

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

$=6 \times (21)^2+\frac{22}{7} \times (\frac{21}{2})^2$

$=2646+346.5$

$=2992.5 \mathrm{~cm}^{2}$

The volume and total surface area of the remaining block are $6835.5\ cm^3$ and 2992.5\ cm^2$ respectively.

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Updated on: 10-Oct-2022

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