The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side.

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