Q. 23

# Find the ratio of the volume of a cube to that of a sphere which will fit inside it.

Let the radius of the sphere be ‘R’ units

And the cube which will fit inside it be of edge ‘a’ units

Explanation: The longest diagonal of the cube that will fit inside the sphere will be the diameter of the of the sphere.

The longest diagonal of cube = the diameter of the sphere

Consider ΔBCD, BDC = 90°

BD = CD = a units (as they are the edges of cube) BC2 = a2 + a2 (putting value of BD and CD)

BC2 = 2a2

BC = √(2a2)

BC = a√2 units eqn1

Now consider ΔABC, ABC = 90°

Here, AB = a units and BC = a√2 units AC2 = a2 + (a√2)2 (putting values of AB and BC)

AC2 = a2 + 2a2

AC2 = 3a2

AC = √(3a2)

AC = a√3 units

Diameter of sphere = D = a√3 units

And we know, D = 2 × R

R = D/2 (put value of D ) Also, Volume of a sphere eqn2

Put value of R in eqn2  Volume of sphere = πa2 cubic units eqn3

Volume of cube = (edge)3

Volume of cube = a3 cubic units eqn4

Ratio of volume of cube to that of sphere  (putting values from eqn3 and eqn4)

Ratio of volume of cube to that of sphere  Ratio of volume of cube to that of sphere is a:π

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