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.

Complete Solution:

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 part

The 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 cube

CSA of hemisphere= 2πr²

Area of the base of the hemisphere = πr², where r is the radius of the hemisphere

Diameter of the hemisphere = Length of the edge of the cube = l

The radius of the hemisphere, r = l/2

The 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 + 2πr² - πr²

= 6l² + πr²

= 6l² + π (l/2)²

= 6l² + πl² / 4

= ¼ l² (π  + 24)

Final Answer

Thus, the surface area of the remaining solid is ¼ l² (π  + 24).