The length of a line measured with a chain was found to be 250 m. What is the true length of the line if the length was measured with a 30 m chain and the chain was 10 cm long?
- A250.23 m
- B250.43 m
- C250.63 m
- D250.83 mCorrect
Solution
Concept
When a chain or tape is of incorrect length, the measured distances are affected and must be corrected using the ratio of the incorrect length to the true length of the chain.
Formula
True Length (L') = Measured Length (L) × (Incorrect Length of Chain (l') / True Length of Chain (l))
Step-by-step solution
1. Identify the given values: Measured length (L) = 250 m, Nominal length of chain (l) = 30 m, Error in chain = +10 cm = +0.10 m. 2. Determine the incorrect length of the chain (l'): l' = 30 m + 0.10 m = 30.10 m. 3. Apply the correction formula: True Length = L × (l' / l). 4. Substitute the values: True Length = 250 × (30.10 / 30) = 250.833 m.
Explanation
The true length of a line is calculated using the relation: True Length = (Measured Length × Incorrect Length of Chain) / True Length of Chain. Here, the measured length is 250 m, the erroneous chain length is 30.10 m (since it is 10 cm too long), and the nominal (true) length of the chain is 30 m.
Why the other options are wrong
A: Calculated using incorrect sign convention or wrong ratio. B: Results from improper arithmetic approximation. C: Arises from subtracting the error instead of adding it for a long chain.
Shortcut
True Length ≈ Measured Length × (1 + Error / Nominal Length) when the error is very small.
Common mistake
Forgetting to add the 10 cm to the total length of the 30 m chain, or subtracting it instead of adding when the chain is too long.
Correct answer: D) 250.83 m