Answer :

Given: a volume of cube increasing at a constant rate


To prove: the increase in its surface area varies inversely as the length of the side


Explanation: Let the length of the side of the cube be ‘a’.


Let V be the volume of the cube,


Then V = a3……..(i)


As per the given criteria the volume is increasing at a uniform rate, then



Now substituting the value from equation (i) in above equation, we get



Now differentiating with respect to t we get




Now let S be the surface area of the cube, then


S = 6a2


Now differentiating surface area with respect to t, we get



Applying the derivatives, we get



Now substituting value from equation (ii) in the above equation we get



Cancelling the like terms we get



Converting this to proportionality, we get



Hence the surface area of the cube with given condition varies inversely as the length of the side of the cube.


Hence Proved


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