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.

Given:

A cubical block of side $7\ cm$ is surmounted by a hemisphere.

To do:

We have to find the greatest diameter the hemisphere can have and the surface area of the solid.

Solution:

Side of the cubical block, $a = 7\ cm$

Longest diagonal of the cubical block $= a\sqrt{2}\ cm=7\sqrt{2}\ cm$

Since the cube is surmounted by a hemisphere,

Therefore the side of the cube should be equal to the diameter of the hemisphere.

Diameter of the sphere$= 7\ cm$

Radius of the sphere, $r = \frac{7}{2}\ cm$

Total surface area of the solid $=$Total surface area of the cube$-$ Inner cross‐section area of the hemisphere $+$ Curved surface area of the hemisphere

$=6a^{2} -\pi r^{2} +2\pi r^{2}$

$=6a^{2} +\pi r^{2}$

$=6\times 7\times 7+\frac{22}{7}\times \frac{7}{2} \times \frac{7}{2}$

$=294+\frac{77}{2}$

$=332.5\ cm^{2}$.

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Updated on: 10-Oct-2022

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