The initial and final temperatures of a water bath are (18 ± 0.5)°C and (40± 0.3)°C. What is the rise in temperature of the bath ?

To find the rise in temperature of the water bath, we need to calculate the difference between the final temperature and the initial temperature, and apply the rules of error propagation.

The initial temperature is given as:

$T_1=(18\pm 0.5)°C$

The final temperature is given as:

$T_2=(40\pm 0.3)°C$

Step 1: Calculate the nominal rise in temperature
The nominal rise in temperature is the difference between the nominal values of the final and initial temperatures:

$\Delta T=T_2-T_1$

$\Delta T=40-18=22°C$

Step 2: Calculate the absolute error in the temperature rise
According to the rules of error analysis, when subtracting (or adding) two quantities, their absolute errors are always added together to obtain the absolute error in the final result:

$\delta \left(\Delta T\right)=\delta T_1+\delta T_2$

Substituting the given absolute errors of 0.5°C and 0.3°C:

$\delta \left(\Delta T\right)=0.5+0.3=0.8°C$

Step 3: State the final result
Combining the nominal value and the propagated error, we express the rise in temperature as:

$\text{Rise in temperature}=(22\pm 0.8)°C$