The sum of the surface areas of a sphere and a cube is given. Show that when the sum of their volumes is least, the diameter of the sphere is equal to the edge of the cube.

Explanation

Surface area of sphere and cube,

S=`4 pi r^2+6 a^2`

edge of cube,

a=`(sqrtS-4 pi r^2)/6 ldots ldots ldots`(1)

Volume,

V=`(4 pi r^3)/3+a^3`

`Rightarrow V=(4 pi r^3)/3+(S-(4 pi r^2)/(6)^(3/2)`

differentiating and equating it to zero

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