Q. 23

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

Answer :

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)

⇒ BC^{2} = a^{2} + a^{2} (putting value of BD and CD)

⇒BC^{2} = 2a^{2}

⇒BC = √(2a^{2})

∴ BC = a√2 units →eqn1

Now consider ΔABC, ∠ABC = 90°

Here, AB = a units and BC = a√2 units

⇒AC^{2} = a^{2} + (a√2)^{2} (putting values of AB and BC)

⇒ AC^{2} = a^{2} + 2a^{2}

⇒AC^{2} = 3a^{2}

⇒AC = √(3a^{2})

∴ 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 = πa^{2} cubic units → eqn3

Volume of cube = (edge)^{3}

∴ Volume of cube = a^{3} 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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