Density and Pressure — Question 9

Question 9

Bounds on a quotient are asymmetric

A mass is bounded by $98\le m\le102\,\mathrm{g}$ and its volume by $49\le V\le51\,\mathrm{cm^3}$. These are hard independent bounds, not standard deviations. The nominal values are $100\,\mathrm{g}$ and $50\,\mathrm{cm^3}$.

1. Find the tight density interval and compare it with the nominal density.

2. Can a material of density $2.10\,\mathrm{g/cm^3}$ satisfy both bounds? Explain why simply adding or averaging the measurement endpoints would fail.

Question 9 - Solution

Bounds on a quotient are asymmetric

(a) Derivation and calculation

$$\rho=m/V,\qquad\frac{\partial\rho}{\partial m}=1/V>0,\quad\frac{\partial\rho}{\partial V}=-m/V^2<0.$$$$\boxed{\rho_{\min}=98/51=1.92157},\quad\boxed{\rho_{\max}=102/49=2.08163}\quad\mathrm{g/cm^3}.$$$$\rho_{\mathrm{nominal}}=2.00000,\quad\Delta\rho_-=-.07843,\quad\Delta\rho_+=+.08163.$$

(b) Result and interpretation

$$2.10>2.08163\Rightarrow\boxed{2.10\,\mathrm{g/cm^3}\ \text{is incompatible}}.$$The tight extrema pair small mass with large volume, and large mass with small volume. Averaging endpoints supplies a central value, not a guaranteed bound.

Physical check

The asymmetric interval arises from the reciprocal dependence on volume, even though the input intervals are symmetric.