---
title: "True length: tape found 2.5 cm too long"
subject: "Surveying I"
chapter: 2
key: "semester_3/surveying_i:tape-825-plus-750-m"
source: https://www.bctengineer.com/q/semester_3/surveying_i/tape-825-plus-750-m
site: BCT Engineering
---

# True length: tape found 2.5 cm too long

Surveying I · Chapter 2: Distance Measurements

- Asked in: 2080 Bhadra, 6 marks
- Video: [Surveying Tape Correction (Too Long) | CELE MSTE](https://www.youtube.com/watch?v=-QwJbLnEE6M) — Tutor Jack Ph
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/tape-825-plus-750-m
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-2

## Question

A line was measured with a tape which was exactly 30 m long at 20°C temperature and 10 kg pull during the commencement of the work, and the length measured under such condition of tape is 825 m. After measuring 825 m, the remaining length measured was 750 m, and the same tape was found 2.5 cm too long at the end of the work. Find the true length of the line measured. Temperature during measurement for both conditions is 30°C, coefficient of linear expansion $\alpha = 3.5\times10^{-6}$/°C, cross-sectional area of tape $a = 0.025$ cm², Young modulus of elasticity $E = 2.1\times10^{6}$ kg/cm² and pull applied during both conditions is 12 kg.

## Short answer

Correct the 750 m part for the tape being 2.5 cm too long (+0.625 m), then apply temperature correction +0.0551 m and pull correction +0.0600 m to the total 1575 m. True length is 1575.740 m.

## Answer

**Reading.** The tape was exactly 30 m (standard) at the start, so the first 825 m has no tape-length error. It was found 2.5 cm too long (30.025 m) at the end, so the remaining 750 m, measured with the longer tape, needs a tape-length correction. The temperature and pull corrections apply to the whole line. No sag is mentioned, so the tape is taken as fully supported.

#### Data
- Standard: $T_0 = 20^\circ$C, $P_0 = 10$ kg; field: $T = 30^\circ$C, $P = 12$ kg
- $\alpha = 3.5\times 10^{-6}$/$^\circ$C, $A = 0.025$ cm$^2$, $E = 2.1\times 10^{6}$ kg/cm$^2$
- Total measured length $= 825 + 750 = 1575$ m

#### 1. Tape length correction (750 m part only)
$$
C_a = \frac{0.025}{30}\times 750 = +0.6250\ \text{m}
$$

#### 2. Temperature correction (whole line)
$$
C_t = \alpha\,(T - T_0)\,L = 3.5\times 10^{-6}\times 10\times 1575 = +0.0551\ \text{m}
$$

#### 3. Pull correction (whole line)
$$
C_p = \frac{(P - P_0)\,L}{A\,E} = \frac{(12 - 10)\times 1575}{0.025\times 2.1\times 10^{6}} = +0.0600\ \text{m}
$$

#### 4. True length
$$
L_{true} = 1575 + 0.6250 + 0.0551 + 0.0600 = 1575.7401\ \text{m}
$$

**Answer: true length of the line $= 1575.740$ m.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why is the tape-length correction applied only to the 750 m part?
   A. The tape was exactly 30 m at the start, so the first 825 m has no length error
   B. The 825 m was on a slope
   C. It is random
   D. The pull was different
   Answer: A. The error (2.5 cm too long) appeared only by the end, affecting the second part. (Key point: Tape length correction scope)

2. C_a for 750 m with tape 2.5 cm too long is
   A. -0.625 m
   B. +0.625 m
   C. +0.0625 m
   D. +6.25 m
   Answer: B. 0.025/30 x 750 = +0.625 m. (Key point: Tape correction value)

3. Why is the temperature correction positive?
   A. Field 30 C is above the standard 20 C, so the tape expanded
   B. It is below
   C. Always positive
   D. Pull
   Answer: A. C_t = alpha (T - T0) L = 3.5e-6 x 10 x 1575 = +0.0551 m. (Key point: Temperature sign)

4. Pull correction for 1575 m (A = 0.025, E = 2.1e6, 2 kg extra) is
   A. +0.0600 m
   B. +0.6 m
   C. +0.006 m
   D. -0.06 m
   Answer: A. 2 x 1575 / (0.025 x 2.1e6) = 0.06 m, giving total 1575.7401 m. (Key point: Pull correction)

## Related in Surveying I

- [True length of 780 m line, tape 0.693 kg](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-30m-780m-line) — Surveying I, chapter 2, asked 2×
- [True length of 1020 m line, 20 m tape](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-20m-1020m-line) — Surveying I, chapter 2, asked 2×
- [True length of 680 m line, 20 m tape](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-20m-680m-line) — Surveying I, chapter 2, asked 3×
- [Tape corrections for 86 m, 10 kg pull](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-20m-86m-pull-10kg) — Surveying I, chapter 2, asked 3×
- [Tape corrections for 86 m, 15 kg pull](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-20m-86m-supports-10m-15kg) — Surveying I, chapter 2, asked 3×
- [Correction per tape length (kg data, 58)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-base-line-correction-per-tape-58) — Surveying I, chapter 2, asked 2×

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