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.

Question

Infinity Learn · Accepted Answer

The figure below of the solid is created as per the given information along with the top view of the solid.From the figure, it’s clear that the surface area of the remaining solid includes TSA of the cube, CSA of the hemisphere, and excludes the base of the hemisphere.The surface area of the remaining solid $$=$$ TSA of cubical part $$+$$ CSA of the hemispherical part $$-$$ Area of the base of the hemispherical partThe remaining area of the solid can be found by using the formulae;TSA of the cube $$=$$ $$6l2$$, where l is the length of the edge of the cubeCSA of $$hemisphere=$$ $$2$$$$π$$$$r2Area$$ of the base of the hemisphere $$=$$ π$$r2$$, where r is the radius of the hemisphereDiameter of the hemisphere $$=$$ Length of the edge of the cube $$=$$ lThe radius of the hemisphere, r $$=$$ $$l/2The$$ surface area of the remaining solid $$=$$ TSA of the cubical part $$+$$ CSA of the hemispherical part $$-$$ Area of the base of the hemispherical $$part=$$ $$6l2$$ $$+$$ $$2$$$$π$$$$r2$$ $$-$$ π$$r2=$$ $$6l2$$ $$+$$ π$$r2=$$ $$6l2$$ $$+$$ π (l/2)2=$$ $$6l2$$ $$+$$ π$$l2$$ $$/$$ $$4=$$ ¼ $$l2$$ $$($$π  $$+$$ $$24)Thus$$, the surface area of the remaining solid is ¼ $$l2$$ $$($$π  $$+$$ $$24).

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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