A 50 m tape is held 2 m out of line. What is the true length?

📝 Detailed Explanation: Correction for Bad Ranging

This problem demonstrates a correction for an error due to bad ranging or misalignment. When the measuring tape is held off the true straight line, the measured length along the tape becomes the hypotenuse of a right-angled triangle. The actual straight-line distance between the points is one of the triangle's legs, which will always be shorter than the hypotenuse.

Applying the Pythagorean Theorem

The situation forms a right-angled triangle where:

• The measured tape length is the hypotenuse = 50 m.
• The distance "out of line" is one leg = 2 m.
• The true length (x) is the other leg.

Using the Pythagorean theorem, which states \(a^2 + b^2 = c^2\):

$$ 50^2 = x^2 + 2^2 $$

$$ 2500 = x^2 + 4 $$

$$ x^2 = 2500 - 4 $$

$$ x^2 = 2496 $$

$$ x = \sqrt{2496} $$

$$ x \approx 49.96 \text{ m} $$

Therefore, the correct, true distance along the straight line is 49.96 m. This type of error is cumulative and always makes the measured length greater than the true length.