Question

Volume The change in the volume \(V=x^{3}\) of a cube when the edge lengths change from \(a\) to \(a+d x\)

Short answer

The change in volume \(dV\) when the edge length changes from \(a\) to \(a + dx\) is approximately \(3a^{2}dx\).

Step-by-step solution

Understand the Volume of a Cube

The volume \(V\) of a cube with edge length \(x\) is given by \(V=x^{3}\). Thus, if the edge length changes from \(a\) to \(a + dx\), the new volume is \((a+dx)^{3}\).

Find the Change in Volume Expression

The change in volume, denoted as \(dV\), is given by the new volume minus the old volume, \(dV = (a + dx)^{3} - a^{3}\). Expand the binomial expression \((a + dx)^{3}\) to \(a^{3} +3a^{2}dx + 3adx^{2} + dx^{3}\).

Simplify the Change in Volume Expression

Simplify the change in volume expression: \(dV = 3a^{2}dx + 3adx^{2} + dx^{3}\), realizing that terms with \(dx^{2}\) and \(dx^{3}\) are insignificant due to \(dx\) being very small. The expression simplifies to: \(dV = 3a^{2}dx\). This tells us that when the edge length of a cube increases by an infinitesimal amount, the volume increases roughly by 3 times the square of the original length times the change.