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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 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=l2

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=6l2

The curved surface area of the hemisphere=2πr2=2π×l22=2π×l24=πl22

The base area of the hemisphere=πr2=π×l22=π×l24=πl24

The surface area of the solid=6l2+πl22-πl24=l24π+24

Hence, the surface area of the remaining solid is l24π+24.


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