Let, #x, S and V# denote the Length of Side, Surface Area and
Volume of a Cube.
Then, #S=6x^2 and V=x^3.#
Hence, #V={(S/6)^(1/2)}^3=(S/6)^(3/2)=S^(3/2)/(6sqrt6)...........(1)#
Diff.ing w.r.t. #S, (dV)/(dS)=1/(6sqrt6)(3/2)S^(1/2)=S^(1/2)/(4sqrt6).#
Recall that, #deltaV~~(dV)/(dS)deltaS= S^(1/2)/(4sqrt6)deltaS.#
#rArr (deltaV)/V=S^(1/2)/(4sqrt6)(deltaS)/V.#
#=S^(1/2)/(4sqrt6)(deltaS)/{S^(3/2)/(6sqrt6)}..........[because, (1)]#
#, i.e., (deltaV)/V~~3/2(deltaS)/S.#
#:." The "%" rise in the volume of cube="100(deltaV)/V,#
#~~(3/2)(100(deltaS)/S)#
#=(1.5")(The "%" rise in the surface area of cube)"#
#=(1.5)(1 %)#
Hence, the % rise in the Volume of Cube #=1.5%#.
Enjoy Maths.!
