The length of a survey line measured with a 30 m chain was found to be 350 m. Calculate the true length of the line if the chain was 12 cm too long.
📝 Detailed Explanation: Correcting for an Incorrect Chain Length
This problem deals with a systematic error in chain surveying. When the measuring instrument (the chain) has a length different from its standard designated length, all measurements taken with it will be incorrect. We must apply a correction to find the true distance.
Understanding the Error
When a measuring chain is longer than its standard length, it leads to an underestimation of the distance. You're using a longer "ruler" than you think, so the number of chain lengths you count to cover a certain distance is fewer than it should be. This results in a measured distance that is shorter than the true distance. Therefore, the true length must be greater than the measured length of 350 m.
Calculation Steps
The fundamental relationship for correction is:
True Length × True Chain Length = Measured Length × Incorrect Chain Length
Or, rearranging to solve for the true length:
$$ \text{True Distance} = \frac{\text{Incorrect Chain Length}}{\text{Correct Chain Length}} \times \text{Measured Distance} $$
Applying the Formula
First, let's list the given information:
- Correct Length of Chain (l): 30 m
- Measured Distance of Line (L'): 350 m
- Error in Chain: 12 cm too long. We must convert this to meters (12 cm = 0.12 m).
- Incorrect Length of Chain (l'): 30 m + 0.12 m = 30.12 m
Now, we can substitute these values into the formula:
$$ L = \frac{l'}{l} \times L' $$
$$ L = \frac{30.12 \text{ m}}{30 \text{ m}} \times 350 \text{ m} $$
$$ L = 1.004 \times 350 \text{ m} $$
$$ L = 351.4 \text{ m} $$
The true length of the survey line is 351.4 m, which corresponds to option (c).