Differentials to find approximate amount of paint

Question

A coat of paint of thickness 0.06 cm is to be applied uniformly to the faces of a cube of edge 32 cm. Use differentials to find the approximate amount of paint required for the job, correct to the nearest cubic centimeter.

Answer 1

Approximation Principle For A Differentiable Function f(x): For a number x in the domain of f(x), take Δx as a small change in the value of x and Δy equal to f(x+Δx)−f(x) as the corresponding change in the value of y equal to f(x). The Approximation Principle then asserts that Δy is equivalent to f'(x)•Δx. That is to say, Δy is very close to f'(x)•Δx for small values of Δx.

First establish a check without differentials by computing [(32 centimeters)² of surface area per face] × (0.06 centimeters of paint thickness over all the surface area of the cube) × (6 faces). This would give 368.64 cubic centimeters of paint required to coat the entire cube at the stated thickness.

Then obtain an approximation by use of differentials by writing V = e³ and D_eV or dV/de = 3e² with Δe here equal to 2(0.06) or 0.12. With e equal to 32 here, 3e² = 3072 and the approximation sought is reached by

D_eVΔe or 3072×0.12 or, again, 368.64 cubic centimeters of paint or 369 cubic centimeters to the nearest centimeter.