Q. 405.0( 1 Vote )

Answer :

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

We know that,

⇒ Surface area of sphere = 4r^{2}

⇒ Surface area of cube = 6l^{2}

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

⇒ S = 4r^{2} + 6l^{2} …… (1)

We also know that,

⇒ Volume of sphere =

⇒ Volume of cube = l^{3}

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

⇒

⇒

At

⇒

⇒

⇒

⇒

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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