A hemispherical depression is cut out from one face of a cubical

Question:

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.

Solution:

It is given that a hemisphere is cut from a cubical box with edge such that

diameter of hemisphere is also l.

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

Surface area of the cubical box with side $l=6 l^{2}$

Let $r$ be the radius of hemisphere

Surface area of the hemisphere $=\pi r^{2}+2 \pi r^{2}$

$=\pi\left(\frac{l}{2}\right)^{2}+2 \pi\left(\frac{l}{2}\right)^{2}\left(\right.$ since $\left.r=\frac{l}{2}\right)$

Surface area of the remaining solid = surface area of cubical box - surface area of hemisphere

$=6 l^{2}-\pi\left(\frac{l}{2}\right)^{2}+2 \pi\left(\frac{l}{2}\right)^{2}$

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

$\frac{=24 l^{2}-\pi l^{2}+2 \pi l^{2}}{4}$

$=\frac{24 l^{2}+\pi l^{2}}{4}$

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

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