The length of a line measured with a 30 m chain is 800.64 m. Afterwards it is found that the chain is 0.05 m too long. The true length of the line is:
801.974 m
Calculating True Length with an Incorrect Survey Chain
When a line is measured using a chain that is not of the correct standard length, the measured length will be inaccurate. To find the true length of the line, a correction must be applied. If the chain is too long, the measured length will be less than the true length. If the chain is too short, the measured length will be more than the true length.
Understanding the Problem
We are given:
- Nominal length of the chain (\(L\)) = 30 m
- Measured length of the line (\(L_m\)) = 800.64 m
- Error in the chain = 0.05 m too long
This means the actual length of the chain (\(L'\)) is the nominal length plus the error.
\[L' = L + \text{Error}\] \[L' = 30 \text{ m} + 0.05 \text{ m}\] \[L' = 30.05 \text{ m}\]So, the chain that was used for measurement was actually 30.05 m long instead of 30 m.
Formula for True Length
The relationship between the true length (\(L_t\)) and the measured length (\(L_m\)) when using an incorrect chain is given by the formula:
\[L_t = L_m \times \frac{L'}{L}\]Where:
- \(L_t\) is the true length of the line.
- \(L_m\) is the measured length of the line.
- \(L'\) is the actual length of the chain.
- \(L\) is the nominal length of the chain.
Step-by-Step Calculation
Now, we can substitute the given values into the formula:
\[L_t = 800.64 \text{ m} \times \frac{30.05 \text{ m}}{30 \text{ m}}\]First, calculate the ratio of the actual chain length to the nominal chain length:
\[\frac{L'}{L} = \frac{30.05}{30}\] \[\frac{L'}{L} = 1.001666...\]Now, multiply the measured length by this ratio:
\[L_t = 800.64 \times 1.001666...\] \[L_t = 801.9744 \text{ m}\]Rounding the result to three decimal places, the true length of the line is approximately 801.974 m.
Comparing with Options
Let's compare our calculated true length with the given options:
| Option | Value | Comparison |
|---|---|---|
| 1 | 801.976 m | Close, but not exact |
| 2 | 799.305 m | Incorrect |
| 3 | 801.974 m | Matches calculated value |
| 4 | 799.307 m | Incorrect |
The calculated true length, 801.974 m, matches Option 3.
This makes sense because the chain was too long (30.05m instead of 30m). When you measure a fixed distance with a chain that is longer than it should be, you will record a smaller number of chain lengths, resulting in a measured distance that is less than the true distance. Therefore, the true length must be greater than the measured length (800.64 m), which is consistent with our result (801.974 m).
Important Questions from Accuracy and Errors
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1. Negative cumulative error
2. Positive cumulative error
3. Compensating error
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