A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.

Solution

Let a be the length of an edge of the cube. Then
a = 7 cm
Greatest diameter of the hemisphere
= Length of an edge of the cube
= 7 cm
Now,
Surface area of the cube
= 6 (edge)²= 6 x 7²= 6 x 49 = 294 cm²Let r be the radius of the hemisphere.
Then, r = $7 over 2 space cm$
Now,
Curbed surface area of hemisphere
$equals space 2 πr squared equals space open parentheses 2 space straight x space 22 over 7 straight x 7 over 2 straight x 7 over 2 close parentheses equals space 77 space cm squared$
And, Base area = $πr squared equals open parentheses 22 over 7 straight x 7 over 2 straight x 7 over 2 close parentheses space cm squared$
$space space space equals space space 77 over 2 space cm squared$
Total surface area
= Surface area of the cube + curved surface area of the hemisphere – base area of the hemisphere
$equals space open parentheses 294 space plus space 77 minus 77 over 2 close parentheses space cm squared equals space open parentheses 294 plus 77 over 2 close parentheses space cm squared equals space left parenthesis 294 plus 38.5 right parenthesis space cm squared equals space 332.5 space cm squared$