The Celsius and Fahrenheit Scales — Question 4

Question 4

An affine display fault is distinct from changing temperature units

Use numerical readings $F=\frac95C+32$ and $T/\mathrm K=C+273.15$; these conversions are exact. A nominal Fahrenheit display reads $35.0$ in a $0.0^\circ\mathrm C$ reference and $215.0$ in a $100.0^\circ\mathrm C$ reference. Model its reading as $D=aC+b$ exactly over this range.

1. Infer $a,b$ and the true Celsius temperature when $D=125.0$.

2. A technician applies the standard Fahrenheit-to-Celsius conversion directly to $D$. Find the systematic Celsius error and the corrected Fahrenheit reading for the unknown bath.

Question 4 - Solution

An affine display fault is distinct from changing temperature units

(a) Method and calculation

$$a=\frac{215-35}{100-0}=1.8,\qquad b=35.$$$$C=\frac{125-35}{1.8}=50.0.$$The gain is correct but the display has an offset of three Fahrenheit degrees.

(b) Interpretation and further calculation

$$C_{\mathrm{naive}}=\frac{125-32}{1.8}=51.6667,\quad C_{\mathrm{naive}}-C=1.6667.$$$$F=1.8(50)+32=122.0=D-3.0.$$Unit conversion cannot repair an unmodeled calibration fault.

Physical check

Both anchor readings are needed to distinguish an offset fault from an incorrect gain under the affine model.