The volume of a cube is increasing at a constant rate. Prove that the increase in surface area varies inversely as the length of the edge of the cube.

Explanation

let the side of a cube be x unit.
`therefore` Volume of cube (V) =`x^(3)`
On differentiating both sides w.r.t. t, we get
`(dV)/(dt) = 3x^(2)(dx)/(dt)=`k [constant]
`rArr (dx)/(dt) = k/(3x^(2))`............(i)
Also, surface area of cube, `S=6x^(2)`
On differentiating w.r.t. t, we get
`(dS)/(dt) = 12x. (dx)/(dt)`
`(dS)/(dt) = 12x. k/(3x^(2))` [using Eq. (i)]
`rArr (dS)/(dt) = (12k)/(3x) = 4(k/x)`
`rArr (dS)/(dt) propto 1/x`
Hence, the surface area of the cube varies inversely as the length of the side.