Question

The initial and final temperatures of water as recorded by an observer are $(38.6 \pm 0.2)^\circ\text{C}$ and $(82.3 \pm 0.3)^\circ\text{C}$. The rise in temperature with proper error limits is

• A. $(43.7 \pm 0.2)^\circ\text{C}$
• B. $(43.7 \pm 0.3)^\circ\text{C}$
• C. $(43.7 \pm 0.1)^\circ\text{C}$
• D. $(43.7 \pm 0.5)^\circ\text{C}$

Hint:

Errors always add up, whether you are finding a sum or a difference.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When calculating the difference between two measured quantities, the nominal values are subtracted, but their absolute errors are added.
Errors always propagate additively in sums and differences to give the maximum possible uncertainty.
Step 2: Key Formula or Approach:
For quantities $A \pm \Delta A$ and $B \pm \Delta B$:
Difference $Z = A - B$.
Error in difference $\Delta Z = \Delta A + \Delta B$.
Step 3: Detailed Explanation:
Initial temperature $T_1 = 38.6^\circ\text{C}$ with error $\Delta T_1 = 0.2^\circ\text{C}$.
Final temperature $T_2 = 82.3^\circ\text{C}$ with error $\Delta T_2 = 0.3^\circ\text{C}$.
Calculate the nominal rise in temperature $\Delta T$: \[ \Delta T = T_2 - T_1 = 82.3 - 38.6 = 43.7^\circ\text{C} \] Calculate the total absolute error $\Delta(\Delta T)$: \[ \Delta(\Delta T) = \Delta T_1 + \Delta T_2 = 0.2 + 0.3 = 0.5^\circ\text{C} \] Therefore, the rise in temperature is reported as $(43.7 \pm 0.5)^\circ\text{C}$.
Step 4: Final Answer:
The rise in temperature is $(43.7 \pm 0.5)^\circ\text{C}$.