A rectangle has sides a = 4 (exact) and b = 5 ±0.4. Find

A rectangle has sides $a = 4$ (exact) and $b = 5\pm0.4$. Find the maximum possible error in the computed area.

Step 1: The area and its differential

With $S = ab$ and $a$ held fixed, the only variable is $b$: \[\frac{dS}{db} = a, \qquad dS = a\,\Delta b = 4\,\Delta b.\]

The nominal area is $S = 4\cdot5 = 20$ square units.

Step 2: Maximum absolute error

The worst case is $\bigl|\Delta b\bigr| = 0.4$: \[\bigl|\Delta S\bigr|_{\max} = a\,\bigl|\Delta b\bigr|_{\max} = 4\cdot0.4 = 1.6 \text{ square units}.\]

Step 3: Relative error

Dividing by $S = ab$, the exact side cancels completely: \[\varepsilon_{\text{rel}} = \frac{\bigl|\Delta S\bigr|}{S} = \frac{a\bigl|\Delta b\bigr|}{ab} = \frac{\bigl|\Delta b\bigr|}{b} = \frac{0.4}{5} = 0.08 = 8\%.\]

So the area inherits the relative error of the uncertain side unchanged — the coefficient is $1$, not $2$, because only one of the two sides is in doubt.

Step 4: Verification

Here the differential is not an approximation: $S$ is a linear function of $b$, so the increment has no remainder at all.

\[S_{\max} = 4\cdot5.4 = 21.6 = 20+1.6, \qquad S_{\min} = 4\cdot4.6 = 18.4 = 20-1.6 \;\;\checkmark\]

\[\text{The error band is exactly symmetric: } \Delta S = 4\,\Delta b \text{ holds identically, not just for small } \Delta b \;\;\checkmark\]

\[\frac{1.6}{20} = \frac{0.4}{5} = 8\% \;\;\checkmark\]