calculus for small changes

Question

When a cubical block of metal is heated, each edge increases by 0.1% per degree increase in temperature. Use differentials to estimate the percentage increase in the surface area and volume of the block per degree increase in temperature.

Answer 1

A cube has the edges equal length so if "a" designates the length of an edge,

Volume = V = a³ and Surface Area = S = 6a²

dV = 3a² da and dS = 12a da

for .1% or .001 unit change, da would then be .001a for each degree change

So dV = 3a²(.001a) = .003a³ and dS = 12a(.001a) = .012a²

In percentage terms,

dV = .3a³% and dS = 1.2a²%

You could check by taking the original length a and you see the volume is a³

Then you could increase 1 degree and volume is (1.001a)3 = 1.003a³

The difference is 1.003a³ - a³ = .003a³ or .3a³% and so on......