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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

Let the side length of the cubical block be x, and let S(x) and V(x) denote the surface area and volume of the cubical block as functions of the side length x. Then we have:

S(x)=6x²

V(x)=x³

The differential form of these two equations is as follows:

dS=12xdx

dV=3x²dx

If instead of infinitesimal quantities we are dealing with merely very small quantities then we obtain the following approximations:

∆S≈12x∆x

∆V≈3x²∆x

Furthermore:

∆S/S≈12x∆x/S -> ∆S/S≈12x∆x/(6x²) -> ∆S/S≈2(∆x/x)

∆V/V≈3x²∆x/V -> ∆V/V≈3x²∆x/x³ -> ∆V/V≈3(∆x/x)

Since (∆x/x)*100=0.1 -> ∆x/x=0.001 it follows that:

(∆S/S)*100≈2(∆x/x)*100=2*0.001*100=0.2

(∆V/V)*100≈3(∆x/x)*100=3*0.001*100=0.3

Therefore, for each degree increase in temperature the side of the cubical block increases by 0.1%, the surface area increases by approximately 0.2%, and the volume increases by approximately 0.3%.