A surveyor needs to determine the distance between two points under three different scenarios:

Recommend the most appropriate method for horizontal distance measurement for each scenario.

During a taping operation, a surveyor uses a $50 \text{ m}$ steel tape that was standardized at $20^\circ \text{C}$ with a pull of $50 \text{ N}$ fully supported.

Explain whether the measured distance recorded in the field book will be too long or too short compared to the true distance under the following conditions:

A surveyor walked a $50 \text{ m}$ course 5 times with the following number of paces: 62, 63, 61, 64, and 62. They then walked an unknown distance $AB$ with the following paces: 145, 148, 146, and 147.

Determine the surveyor's pace factor and the approximate distance of line $AB$.

A $100 \text{ m}$ steel tape is standardized at $20^\circ \text{C}$. A line is measured with this tape and found to be $850.25 \text{ m}$ at an average field temperature of $32^\circ \text{C}$. The coefficient of linear expansion for steel is $\alpha = 0.0000116 / ^\circ \text{C}$.

Calculate the total temperature correction and the corrected length of the line.

A $50 \text{ m}$ tape weighing $0.03 \text{ kg/m}$ (total mass $1.5 \text{ kg}$) is standardized under a pull of $5 \text{ kg}$. The cross-sectional area is $0.04 \text{ cm}^2$ and $E = 2.1 \times 10^6 \text{ kg/cm}^2$. During measurement, the tape is supported only at its ends ($0 \text{ m}$ and $50 \text{ m}$) and a pull of $8 \text{ kg}$ is applied.

Calculate the pull correction ($C_p$) and the sag correction ($C_s$) for one tape length. Determine the effective length of the tape under these conditions.

Using the data from Problem 3 ($L=50\text{m}$, $W=1.5\text{kg}$, $P_s=5\text{kg}$, $A=0.04\text{cm}^2$, $E=2.1\times 10^6\text{ kg/cm}^2$), determine the Normal Tension ($P_n$) required to make the elongation from pull exactly counteract the shortening from sag.

A transit is set up over point A. A stadia rod is held vertically at point B. The transit has a stadia interval factor ($K$) of 100 and a stadia constant ($C$) of 0.30 m. The upper stadia hair reading is 2.450 m and the lower stadia hair reading is 1.150 m. The line of sight is horizontal.

Determine the horizontal distance from the transit to the rod.