True length: tape found 2.5 cm too long

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.