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 +-0.3 m`
D. `1.4 m+-0.2m`

Explanation

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

### Step 1: Identify the values and uncertainties
We have:
- \(A = 1.0 \, m\) with an uncertainty of \(\Delta A = 0.2 \, m\)
- \(B = 2.0 \, m\) with an uncertainty of \(\Delta B = 0.2 \, m\)

### Step 2: Calculate the value of \(\sqrt{AB}\)
We need to calculate:
\[ x = \sqrt{AB} = \sqrt{(1.0 \, m)(2.0 \, m)} = \sqrt{2.0 \, m^2} = 1.414 \, m \]
For simplicity, we can round this to:
\[ x \approx 1.41 \, m \]

### Step 3: Calculate the relative uncertainty in \(x\)
The formula for the relative uncertainty in a product or a power is given by:
\[ \frac{\Delta x}{x} = \frac{1}{2} \left(\frac{\Delta A}{A} + \frac{\Delta B}{B}\right) \]
Substituting the values:
\[ \frac{\Delta A}{A} = \frac{0.2 \, m}{1.0 \, m} = 0.2 \]
\[ \frac{\Delta B}{B} = \frac{0.2 \, m}{2.0 \, m} = 0.1 \]
Now substituting these into the equation:
\[ \frac{\Delta x}{x} = \frac{1}{2} (0.2 + 0.1) = \frac{1}{2} \cdot 0.3 = 0.15 \]

### Step 4: Calculate the absolute uncertainty \(\Delta x\)
Now we can find \(\Delta x\) using:
\[ \Delta x = x \cdot \frac{\Delta x}{x} = 1.414 \, m \cdot 0.15 \approx 0.212 \, m \]
Rounding this gives:
\[ \Delta x \approx 0.21 \, m \]

### Step 5: Report the final result
Thus, the final result for \(\sqrt{AB}\) is:
\[ \sqrt{AB} = 1.41 \, m \pm 0.21 \, m \]