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.

Answer 1

Step 1: Find the radius of the hemisphere

The diameter of the hemisphere is equal to the edge of the cube,

Thus, diameter = side of cube =$l$

So, the radius of the hemisphere$=\frac{l}{2}$

Step 2: Find the surface area of the remaining solid

The surface area of solid = Total surface area of the cube + Curved surface area of the hemisphere – Base area of the hemisphere

The total surface area of the cube$=6l^2$

The curved surface area of the hemisphere$=2{\mathrm{\pi r}}^2=2\mathrm{\pi }\times {\left(\frac{\mathrm{l}}{2}\right)}^2=2\mathrm{\pi }\times \frac{{\mathrm{l}}^2}{4}=\frac{{\mathrm{\pi l}}^2}{2}$

The base area of the hemisphere$={\mathrm{\pi r}}^2=\mathrm{\pi }\times {\left(\frac{\mathrm{l}}{2}\right)}^2=\mathrm{\pi }\times \frac{{\mathrm{l}}^2}{4}=\frac{{\mathrm{\pi l}}^2}{4}$

The surface area of the solid$=6l^2+\frac{{\mathrm{\pi l}}^2}{2}-\frac{{\mathrm{\pi l}}^2}{4}=\frac{l^2}{4}\left(\mathrm{\pi }+24\right)$

Hence, the surface area of the remaining solid is $\frac{l^2}{4}\left(\mathrm{\pi }+24\right)$.