Question

The length of a line measured with a 30 m chain was found to be 500 m. Calculate the true length of the line if chain was 5 cm too long.

• A. \(499.17 \text{ m}\)
• B. \(500 \text{ m}\)
• C. \(500.83 \text{ m}\)
• D. \(501.66 \text{ m}\)

Hint:

If the chain is too long, the true distance is always greater than the measured distance. If the chain is too short, the true distance becomes smaller than the measured distance.

Answer 1

The Correct Option is C

Concept: In chain surveying, if the actual length of the chain differs from its nominal length, correction must be applied to the measured distance. When the chain is too long, the measured distance becomes smaller than the true distance. This happens because each chain length laid on the ground is actually longer than assumed. The correction formula is: \[ \text{True Length} = \text{Measured Length} \times \frac{\text{Actual Chain Length}}{\text{Nominal Chain Length}} \] where,
• Nominal length = Standard length of chain
• Actual length = Real length of chain

Step 1: Write the given data carefully. Measured length of line: \[ L_m = 500 \text{ m} \] Nominal length of chain: \[ L = 30 \text{ m} \] The chain is \(5 \text{ cm}\) too long. Convert centimetres into metres: \[ 5 \text{ cm} = \frac{5}{100} = 0.05 \text{ m} \] Therefore, actual chain length becomes: \[ L_a = 30 + 0.05 \] \[ L_a = 30.05 \text{ m} \]

Step 2: Apply the correction formula. Using: \[ \text{True Length} = \text{Measured Length} \times \frac{\text{Actual Length}}{\text{Nominal Length}} \] Substituting values: \[ \text{True Length} = 500 \times \frac{30.05}{30} \]

Step 3: Simplify the fraction. \[ \frac{30.05}{30} = 1.0016667 \] Now multiply: \[ 500 \times 1.0016667 \] \[ = 500.83335 \] Thus, \[ \boxed{\text{True Length} \approx 500.83 \text{ m}} \]

Step 4: Select the correct option. Therefore, the correct answer is: \[ \boxed{(C)\ 500.83 \text{ m}} \]