Multivariable and Vector Calculus · Constrained optimization and Lagrange multipliers
A closed rectangular box has measured edge lengths x=8 cm, y=5 cm, and z=3 cm
Problem
A closed rectangular box has measured edge lengths \(x=8\,\mathrm{cm}\), \(y=5\,\mathrm{cm}\), and \(z=3\,\mathrm{cm}\). Each measurement may be in error by at most \(0.02\,\mathrm{cm}\). Let \(V=xyz\) be the volume and let \[ S=2(xy+xz+yz) \] be the surface area. (a) Compute the measured values of \(V\) and \(S\). (b) Write the total differentials \(dV\) and \(dS\) in terms of \(dx\), \(dy\), and \(dz\), evaluated at the measured dimensions. (c) Use differentials to estimate the maximum possible error in \(V\) and the maximum possible error in \(S\). (Replace each of \(\lvert dx\rvert\), \(\lvert dy\rvert\), \(\lvert dz\rvert\) by \(0.02\) and use the triangle inequality.) (d) Estimate the maximum possible relative error in \(V\), that is \(\lvert dV\rvert/V\), as a reduced fraction.
Hint
The worst-case error bound from a differential is the sum of the absolute values of the coefficients times the maximum increment in each variable.
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