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.

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Solution

## Step 1: Find the radius of the hemisphereThe 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 solidThe surface area of solid = Total surface area of the cube + Curved surface area of the hemisphere – Base area of the hemisphereThe total surface area of the cube$=6{l}^{2}$The curved surface area of the hemisphere$=2{\mathrm{\pi r}}^{2}=2\mathrm{\pi }×{\left(\frac{\mathrm{l}}{2}\right)}^{2}=2\mathrm{\pi }×\frac{{\mathrm{l}}^{2}}{4}=\frac{{\mathrm{\pi l}}^{2}}{2}$The base area of the hemisphere$={\mathrm{\pi r}}^{2}=\mathrm{\pi }×{\left(\frac{\mathrm{l}}{2}\right)}^{2}=\mathrm{\pi }×\frac{{\mathrm{l}}^{2}}{4}=\frac{{\mathrm{\pi l}}^{2}}{4}$The surface area of the solid$=6{l}^{2}+\frac{{\mathrm{\pi l}}^{2}}{2}-\frac{{\mathrm{\pi l}}^{2}}{4}=\frac{{l}^{2}}{4}\left(\mathrm{\pi }+24\right)$Hence, the surface area of the remaining solid is $\frac{{l}^{2}}{4}\left(\mathrm{\pi }+24\right)$.

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