A cube is being manufactured with a side length of 78 cm and a maximum possible error of 0.031 cm for the sides. Find the approximate error and the approximate relative error in the volume of the cube. Give the relative error as a percent.

The differential is $LaTeX: \displaystyle dV = 3 s^{2}$ . Evaluating at $LaTeX: \displaystyle s = 78$ and $LaTeX: \displaystyle ds = 0.031$ gives $LaTeX: \displaystyle 565.812$ . The relative error is given by $LaTeX: \displaystyle \frac{dV}{V} = \frac{3 s^{2}}{s^{3}}\,ds=\frac{3}{s}\,ds$ . Evaluating gives $LaTeX: \displaystyle 0.00119 = 0.119$ %