Question

A 100 m tape is held 1 m out of line. What is the true length of tape?

• A. 99.990
• B. 99.998
• C. 99.995
• D. 99.992

Hint:

Out-of-line alignment errors always make the measured distance longer than the true distance.
Therefore, the alignment correction is always negative, and the true distance is always shorter than the nominal length.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
When measuring distances with a tape, errors can occur if the tape is not held in a straight line between the two stations.
This is known as an out-of-line or alignment error.
This misalignment makes the measured distance appear longer than the actual straight-line distance, requiring a negative correction.
Key Formula or Approach:
The correction for alignment error (\(C_a\)) is mathematically identical to the slope correction: \[ C_a = -\frac{h^2}{2L} \] where:
- \(h\) is the distance the tape is held out of line (\(\text{m}\)).
- \(L\) is the nominal length of the tape (\(\text{m}\)).
The true straight-line length (\(L_{\text{true}}\)) is calculated as: \[ L_{\text{true}} = L + C_a \]

Step 2: Detailed Explanation:
Let's list the given parameters:
- Nominal length of the tape (\(L\)) = \(100 \text{ m}\)
- Distance out of line (\(h\)) = \(1 \text{ m}\)
First, calculate the alignment correction: \[ C_a = -\frac{1^2}{2 \times 100} \] \[ C_a = -\frac{1}{200} = -0.005 \text{ m} \] Now, calculate the true straight-line length of the tape: \[ L_{\text{true}} = L + C_a \] \[ L_{\text{true}} = 100 \text{ m} - 0.005 \text{ m} = 99.995 \text{ m} \]

Step 3: Final Answer:
The true length of the tape is 99.995 m.