Question:
The surface areas of a sphere and cube are equal. Show that the ratio of the volumes of the sphere and cube is √6:√π
Answer:
Surface area of sphere = 4*π*r2 where r is the radius of sphere.
Surface area of cube = 6a2 where a is the lenght of each edge of cube.
Given in the question,
Surface area of sphere = Surface area of cube
=> 6a2 = 4*π*r2
=> a2 = (4/6)*π*r2
=> a2 = (2/3)*π*r2 (when we divide 4 and 6 by 2)
=> a = (√(2/3)*π)*r
Now,
Volume of sphere/volume of cube
=> ((4/3)*π*r3 )/a3
=> ((4/3)*π*r3 )/{(2/3 * pie*r2 )*(√2*√π/√3 *r)} (By putting value of a and a2)
=>(4*π*r3 *3*√3)/(2*3*π*r2 *√2*√π*r)
=>(4*π*3*√3)/(2*3*π*√2*√π)
=>(2*2*π*3*√3)/(2*3*π*√2*√π)
=>(2*π*3*√3)/(3*π*√2*√π)
=>(√2*√2*π*3*√3)/(3*π*√2*√π) (2=√2*√2)
=>(√2*√3)/√π
=> √6/√π
So ratio is √6:√π
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