You measure two quantities as `A=1.0m+-0.2m`,`B=2.0m+-0.2m`. We should report correct value for `sqrt(AB)` as

Options

A. `1.4m+-0.4m`
B. `1.41m+-0.15m`
C. `1.4m+-0.3m`
D. `1.4m+-0.2m`

Explanation

To solve the problem, we need to find the value of \( \sqrt{AB} \) given the measurements of \( A \) and \( B \) along with their uncertainties.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- \( A = 1.0 \, \text{m} \pm 0.2 \, \text{m} \)
- \( B = 2.0 \, \text{m} \pm 0.2 \, \text{m} \)

2. **Calculate the Product \( AB \):**
- First, calculate \( AB \):
\[ AB = (1.0 \, \text{m}) \times (2.0 \, \text{m}) = 2.0 \, \text{m}^2 \]

3. **Calculate the Square Root \( \sqrt{AB} \):**
- Now, calculate \( \sqrt{AB} \):
\[ \sqrt{AB} = \sqrt{2.0 \, \text{m}^2} = 1.414 \, \text{m} \]

4. **Determine the Significant Figures:**
- Both \( A \) and \( B \) have 2 significant figures, so \( \sqrt{AB} \) should also be reported with 2 significant figures:
\[ \sqrt{AB} \approx 1.4 \, \text{m} \]

5. **Calculate the Uncertainty in \( \sqrt{AB} \):**
- The formula for the relative uncertainty in a product is:
\[ \frac{\Delta y}{y} = \frac{1}{2} \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right) \]
- Here, \( \Delta A = 0.2 \, \text{m} \) and \( \Delta B = 0.2 \, \text{m} \).
- Substitute the values:
\[ \frac{\Delta y}{1.414} = \frac{1}{2} \left( \frac{0.2}{1.0} + \frac{0.2}{2.0} \right) \]
- Calculate the fractions:
\[ \frac{0.2}{1.0} = 0.2 \quad \text{and} \quad \frac{0.2}{2.0} = 0.1 \]
- Thus:
\[ \frac{\Delta y}{1.414} = \frac{1}{2} (0.2 + 0.1) = \frac{1}{2} (0.3) = 0.15 \]
- Now, calculate \( \Delta y \):
\[ \Delta y = 1.414 \times 0.15 \approx 0.2121 \, \text{m} \]

6. **Round the Uncertainty:**
- The uncertainty should be reported with 1 significant figure (the least precise measurement):
\[ \Delta y \approx 0.2 \, \text{m} \]

7. **Final Result:**
- Combine the calculated value and its uncertainty:
\[ \sqrt{AB} = 1.4 \pm 0.2 \, \text{m} \]

### Final Answer:
\[ \sqrt{AB} = 1.4 \pm 0.2 \, \text{m} \]