The largest volume of a cube that can be enclosed in a sphere of diameter 2cm is (in cm3)
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This is the diagrammatic representation.
The cube with largest volume inside a sphere should fulfil following conditions (which can be found by double differentiation):
The point of concurrence of diagonals of cube should be the centre of the sphere.The plane separating the cube in half should also separate the sphere in half. i.e. 2 faces of cube should be parallel and 4 perpendicular to the plane of diameter of sphere.
Such a cube has diagonal lengths (DF, AG, EC & HB)=diameter of sphere=2 units.
Let the sides of the cube (FG, GC, CB, BF, etc.) be 'a'.
Hence facial diagonal of the cube (FC, BG, FH, etc.) will be √2a (Pythagoras' Theorem).
Applying Pythagoras' theorem for △DFC:
FC²+DC²=FD²
⇒(√2a)²+a²=2²
⇒3a²=4
⇒a=2/√3 units
Volume of cube= a^3
=(2/√3)^3
=8/3√3 cubic units. (~1.5396 cubic units)
The cube with largest volume inside a sphere should fulfil following conditions (which can be found by double differentiation):
The point of concurrence of diagonals of cube should be the centre of the sphere.The plane separating the cube in half should also separate the sphere in half. i.e. 2 faces of cube should be parallel and 4 perpendicular to the plane of diameter of sphere.
Such a cube has diagonal lengths (DF, AG, EC & HB)=diameter of sphere=2 units.
Let the sides of the cube (FG, GC, CB, BF, etc.) be 'a'.
Hence facial diagonal of the cube (FC, BG, FH, etc.) will be √2a (Pythagoras' Theorem).
Applying Pythagoras' theorem for △DFC:
FC²+DC²=FD²
⇒(√2a)²+a²=2²
⇒3a²=4
⇒a=2/√3 units
Volume of cube= a^3
=(2/√3)^3
=8/3√3 cubic units. (~1.5396 cubic units)
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