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Question

You measure two quantities as $$A = 1.0 m \pm 0.2 m$$, $$B = 2.0 m \pm 0.2 m$$. We should report correct value for $$\sqrt{AB}$$ as


A
1.4m±0.4m
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B
1.41m±0.15m
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C
1.4m±0.3m
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D
1.4m±0.2m
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Solution

The correct option is D $$1.4 m \pm 0.2 m$$
Here, $$A = 1.0 m \pm 0.2 m$$, $$B = 2.0 m \pm 0.2 m$$
$$AB = (1.0 m)(2.0 m) = 2.0 m^2$$
$$\sqrt{AB} = \sqrt{2.0m} = 1.414 m$$

Rounding off to two significant figures, we get
$$\sqrt{AB} = 1.4 m$$
$$\displaystyle \frac{\Delta AB}{AB} = \frac{1}{2} \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right) = \frac{1}{2} \left( \frac{0.2}{1.0} + \frac{0.2}{2.0} \right) = \frac{0.3}{2}$$
$$\displaystyle \Delta \sqrt{AB} = \frac{0.3}{2} \times \sqrt{AB} = \frac{0.3}{2} \times 1.414 = 0.212 m$$

Rounding off to one significant figure, we get
$$\Delta \sqrt{AB} =0.2m$$
The correct value for $$\sqrt{AB}$$ is $$1.4 m \pm 0.2 m$$.

Physics

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