Worked exercise — The halved lead cube

Problem

You have a lead cube with edge $\ell = 1\;\text{m}$. A friend asks you for half of the cube. So you melt down half of your lead and make a new cube out of it. What is the edge of this new cube?

Solution

Step 1 — Starting volume. The volume of the original cube is

$V = \ell^3 = 1\;\text{m}^3$

Step 2 — Half the volume. Half of that volume is $V' = 0{,}5\;\text{m}^3$. The new cube has an edge $\ell'$ such that $\ell'^3 = 0{,}5$, so:

$\ev{\ell' = \sqrt[3]{0{,}5} \approx 0{,}794\;\text{m}}$

Step 3 — Interpretation. The result is not $\ell/2 = 0{,}5\;\text{m}$! Volume scales as the cube of the linear dimension: to halve a volume, the edge only needs to be reduced to about $79\%$, not to $50\%$. The same scaling law explains why a skyscraper twice as tall as another has eight times more interior space, and why ants (small) can lift proportionally far more weight than elephants (large).

Links

Topics: Physical quantities and measurement Concepts: Scientific notation and orders of magnitude Skills: Dimensional analysis