Question

A thermometer has it's stem marked in millimeter instead of degree Celsius.The lower fixed point is 30mm and the upper fixed point is 180mm. Calculate the temperature in degree Celsius when the thermometer reads 45mm.

Answer 1

Let's find the transfer function: the dependence of liquid height on the temperature $h(T)$. Assuming this dependence to be linear, have:

$$h(T) = kT +b$$

where $k,b$ are some coeffitients. Using the given information, we can write:

$$h(0) = k\cdot 0\degree C + b = 30mm\space \space \space\text{(lower fixed point for Celsius is T = 0\degree C)}\\ h(100) = k\cdot 100\degree C + b =180mm\space \space \space\text{(upper fixed point for Celsius is T = 100\degree C)}$$

From the first equation find:

$$b = 30mm$$

Substituting this into the second equation, find:

$$100\degree C \cdot k + 30mm = 180mm\\ k = \dfrac{150mm}{100\degree C} = 1.5\dfrac{mm}{\degree C}$$

Finally, obtain:

$$h(T) = 1.5\dfrac{mm}{\degree C}\cdot T + 30mm$$

Substituting $h(T) = 45mm$ and solving for $T$, obtain:

$$45mm = 1.5\dfrac{mm}{\degree C}\cdot T + 30mm\\ T = \dfrac{15mm}{1.5\dfrac{mm}{\degree C}} = 10\degree C$$

Answer. $10\degree C$.