Question

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

Answer 1

Let the side of a cube be x unit.

Volume of cube = V = x³

\(\frac{dV}{dt} = 3x^2 \frac{dx}{dt} = k\)  (constant)

\(\frac{dx}{dt} = \frac k{3x^2}\)

Surface area = S = 6x²

\(\frac{dS}{dt} = 12x \frac{dx}{dt} \)

\(\frac{dS}{dt} = 12x \frac{k}{3x^2} = 4(\frac k x)\)

Hence, the surface area of the cube varies inversely as length of side.