A 30 m tape was tested before a survey and found to have a length of 29.93 m. If the length of a line measured with this tape is 270 m, find the true length of the line.

📏 Core Concept: Correction for Incorrect Tape Length

When a measuring tape has a length different from its standard designated length, all measurements taken with it will be incorrect. This introduces a systematic error that needs to be corrected.

• If a tape is too short (like in this problem), it will 'over-measure' the distance. The recorded measurement will be greater than the true distance.
• If a tape is too long, it will 'under-measure' the distance. The recorded measurement will be less than the true distance.

🔬 Calculation and Formula

The fundamental principle for this correction is that the true length multiplied by the true tape length is equal to the measured (incorrect) length multiplied by the incorrect tape length. The formula is:

$$ L \times D = L' \times D' $$

Where:

• \(L\) = Correct (Standard) Length of the Tape = 30 m
• \(D\) = True Length of the Line = ?
• \(L'\) = Incorrect Length of the Tape = 29.93 m
• \(D'\) = Measured (Incorrect) Length of the Line = 270 m

Now, we can substitute the values and solve for the True Distance (D):

$$ 30 \times D = 29.93 \times 270 $$

$$ D = \frac{29.93 \times 270}{30} $$

$$ D = 269.37 \, \text{m} $$

💡 Study Tip

A quick way to check your answer is to reason about the error. Since the tape was shorter than 30 m, it means the surveyor recorded more "tape lengths" than were actually there. Therefore, the true length of the line must be less than the measured 270 m. This helps you immediately eliminate options A, B, and D.