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Question
You measure two quantities as A = 1.0 m ± 0.2 m, B = 2.0 m ± 0.2 m. We should report correct value for `sqrt(AB)` as ______.
Options
1.4 m ± 0.4 m
1.41 m ± 0.15 m
1.4 m ± 0.3 m
1.4 m ± 0.2 m
Solution
You measure two quantities as A = 1.0 m ± 0.2 m, B = 2.0 m ± 0.2 m. We should report correct value for `sqrt(AB)` as 1.4 m ± 0.2 m.
Explanation:
When we take significant figures of measured quantities, we also include zeros in significant figures. Here quantities A and b are the measuring quantities thus the number of significant figures in 1.0 m and 2.0 m is two.
`sqrt(AB) = sqrt(1.0 xx 2.0)`
= `sqrt(2)`
= 1.414 m
Here the minimal significant figure is two in the significant figures, 1.0 m and 2.0 n. Thus,
`x = sqrt(AB)` = 1.4
`(Δx)/x = 1/2 [(ΔA)/A + (ΔB)/B]`
= `1/2 0.2 [1/1.0 + 1/2.0]`
= `0.1 [(2.0 + 1.0)/(1.0 + 2.0)]`
`(Δx)/1.4 = 0.1 (3.0/2.0)`
= 2.1 m
`Δx` = 0.2 m rounding to one place of decimal.
∴ `sqrt(AB) = (1.4 +- 0.2)` m.
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