Problem I. Errors in Chain Surveying

a$-$ A line was measured with a steel tape which was exactly $30m$ long at $18C$ and found to be $452\mathrm{.}343m.$ The temperature during measurement was $32C.$

Given: coefficient of expansion of the tape $1\mathrm{.}16**{10}^{-5}/$unit length $\frac{?}{?}C.$

Identify the type of the error and find the true length of the line.

b$-$ A $50m$ tape is suspended between the ends under a pull of $150N.$ The mass of the tape is $1\mathrm{.}52$ kilograms.

Identify the type of the error and find the true length of the line

c$-$ A line was measured with a steel tape which was exactly $30ft$ at $18F$ and a pull of $50lb$ and the measured length was $459\mathrm{.}242ft.$ Temperature during measurement was $28F$ and the pull applied was $100lb.$ The tape was uniformly supported during the measurement.

Given: cross$-$sectional area of the tape was $0\mathrm{.}02i{n}^2,$ coefficient of expansion ${6\mathrm{.}45}^{**}\frac{{10}^{-6}}{?}$ unit length $\frac{?}{?}F,$ and modulus of elasticity of steel $E_{\text{s}\text{t}\text{e}\text{e}\text{l}}={29}^{**}{10}^{{?}^{{?}^{?}}}6l\frac{b}{i}{n}^2$

Identify the different types of errors and find the true length of the line.