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

Explanation

To find the correct value for \(\sqrt{AB}\) given the measurements \(A = 1.0 \, \text{m} \pm 0.2 \, \text{m}\) and \(B = 2.0 \, \text{m} \pm 0.2 \, \text{m}\), we will follow these steps:

### Step 1: Calculate the value of \(\sqrt{AB}\)

We start by calculating \(AB\):
\[ AB = A \times B = (1.0 \, \text{m}) \times (2.0 \, \text{m}) = 2.0 \, \text{m}^2 \]

Now, we take the square root:
\[ \sqrt{AB} = \sqrt{2.0 \, \text{m}^2} = 1.414 \, \text{m} \]

### Step 2: Determine the number of significant figures

Next, we need to determine how many significant figures to keep in our result. The values of \(A\) and \(B\) both have 2 significant figures (1.0 and 2.0). Therefore, \(\sqrt{AB}\) should also be reported with 2 significant figures.

### Step 3: Round the result

The calculated value \(1.414 \, \text{m}\) needs to be rounded to 2 significant figures. Since the first two digits are 1 and 4, and the next digit is 1 (which is less than 5), we round down:
\[ \sqrt{AB} \approx 1.4 \, \text{m} \]

### Step 4: Calculate the uncertainty in \(\sqrt{AB}\)

To find the uncertainty \(\Delta y\) in \(\sqrt{AB}\), we use the formula for relative error:
\[ \frac{\Delta y}{y} = \frac{1}{2} \left(\frac{\Delta A}{A} + \frac{\Delta B}{B}\right) \]

Substituting the values:
- \(\Delta A = 0.2 \, \text{m}\)
- \(A = 1.0 \, \text{m}\)
- \(\Delta B = 0.2 \, \text{m}\)
- \(B = 2.0 \, \text{m}\)

Calculating the relative error:
\[ \frac{\Delta y}{1.414} = \frac{1}{2} \left(\frac{0.2}{1.0} + \frac{0.2}{2.0}\right) = \frac{1}{2} \left(0.2 + 0.1\right) = \frac{1}{2} \times 0.3 = 0.15 \]

Now, we find \(\Delta y\):
\[ \Delta y = 1.414 \times 0.15 \approx 0.2121 \, \text{m} \]

### Step 5: Round the uncertainty

The uncertainty \(0.2121 \, \text{m}\) should also be reported with 1 significant figure (since the least precise measurement, \(0.2 \, \text{m}\), has 1 significant figure). Therefore, we round \(0.2121\) to \(0.2 \, \text{m}\).

### Final Result

Combining the value and uncertainty, we report:
\[ \sqrt{AB} = 1.4 \, \text{m} \pm 0.2 \, \text{m} \]