Q. 40 5.0( 1 Vote )

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.

Answer :

Let us assume radius of sphere be ‘r’ and length of side of cube is ‘l’

We know that,

Surface area of sphere = 4r2

Surface area of cube = 6l2

According the problem, the sum of surface areas of a sphere and cube is known. Let us assume the sum be S

S = 4r2 + 6l2 …… (1)

We also know that,

Volume of sphere =

Volume of cube = l3

We need the sum of volumes to be least. Let us assume the sum of volumes be V

From (1)

We assume V as a function of r.

For maxima and minima,

Differentiating V again

At r = 0


We get the sum of values least for .

We know that diameter(d) is twice of radius. So,

d = l

Thus proved.

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