# A hemispherical depression is cut out from

Question:

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

Solution:

We have to find the remaining volume and surface area of a cubical box when a hemisphere is cut out from it.

Edge length of cube $(a)=21 \mathrm{~cm}$

Radius of hemisphere $(r)=10.5 \mathrm{~cm}$

Therefore volume of the remaining block,

$=$ Volume of box $-$ Volume of hemisphere

So,

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

$=(21)^{3}-\frac{2}{3}\left(\frac{22}{7}\right)\left(\frac{21}{2}\right)^{3}$

$=(9261-2425.5) \mathrm{cm}^{3}$

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

So, remaining surface area of the box,

$=$ Surface area of box - Area of base of hemisphere + Curved surface area of hemsphere

Therefore,

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

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

Put the values to get the remaining surface area of the box,

$=\left[6(441)+\frac{22}{7}\left(\frac{21}{2}\right)^{2}\right] \mathrm{cm}^{2}$

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