The lengths `l_(1)` and `l_(2)` of two rods are recorded as `(25.2+0.1)` cm and `(16.8+0.1)` cm. To find the sum of the lengths of the rods.

Options

A. The error in the measuremnet is zero
B. The error in the measurement is 0.2
C. The total length of the rods is (42.0+0.0) cm
D. The total length of the two rods is (42.0 + 0.2)cm

Answer

To solve the problem of finding the sum of the lengths of two rods along with their uncertainties, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the lengths and uncertainties**:
- The length of the first rod, \( l_1 = 25.2 \, \text{cm} \) with an uncertainty of \( \Delta l_1 = 0.1 \, \text{cm} \).
- The length of the second rod, \( l_2 = 16.8 \, \text{cm} \) with an uncertainty of \( \Delta l_2 = 0.1 \, \text{cm} \).

2. **Calculate the sum of the lengths**:
\[ l_{\text{total}} = l_1 + l_2 = 25.2 \, \text{cm} + 16.8 \, \text{cm} \]
\[ l_{\text{total}} = 42.0 \, \text{cm} \]

3. **Calculate the total uncertainty**:
- When adding measurements, the total uncertainty is the sum of the individual uncertainties:
\[ \Delta l_{\text{total}} = \Delta l_1 + \Delta l_2 = 0.1 \, \text{cm} + 0.1 \, \text{cm} \]
\[ \Delta l_{\text{total}} = 0.2 \, \text{cm} \]

4. **Combine the results**:
- The final result for the sum of the lengths of the rods, including the uncertainty, is:
\[ l_{\text{total}} = 42.0 \pm 0.2 \, \text{cm} \]

### Final Answer:
The sum of the lengths of the rods is \( 42.0 \pm 0.2 \, \text{cm} \).

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