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.


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




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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