Question

A hemispherical depression is cut out from one face of a cubical wooden block of edge $21cm$, 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.

Answer 1

Given,

Length of edge of the cube $a=21cm$

diameter of hemisphere $d=a=21cm$

hence, radius of the hemisphere $r=\frac{d}{2}=\frac{21}{2}=10.5cm$

Surface area of cube$=6a^2$

Curved surface area of hemisphere $=2\pi r^2$

Area of base of hemisphere$=\pi r^2$

$\text{Total surface area of remaining block = surface area of cube + surface area of hemisphere - area of base of hemisphere}$

$=6a^2+2\pi r^2-\pi r^2$

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

$=(6\times (21{)}^2+\frac{22}{7}\times (10.5{)}^2)c{m}^2$

$=(6\times 441+\frac{22}{7}\times 110.25)c{m}^2$

$=(2646+346.5)c{m}^2$

$\text{Total surface area of remaining block}=2992.5c{m}^2$

Volume of the cube$=a^3$

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

Hence,

$\text{Volume of remaining block = volume of cube - volume of hemisphere}$

$=a^3-\frac{2}{3}\pi r^3$

$=({21}^3-\frac{2}{3}\times \frac{22}{7}\times (10.5{)}^3)c{m}^3$

$=(9261-\frac{2}{3}\times \frac{22}{7}\times 1157.625)c{m}^3$

$=(9261-2425.5)c{m}^3$

$\text{Volume of the remaining block}=6835.5c{m}^3$