# Tacheometric, Curve and Hydrographic Surveying

 Question 1
A delivery agent is at a location R. To deliver the order, she is instructed to travel to location $P$ along straight-line paths of $\mathrm{RC}, \mathrm{CA}, \mathrm{AB}$ and $\mathrm{BP}$ of $5 \mathrm{~km}$ each. The direction of each path is given in the table below as whole circle bearings. Assume that the latitude (L) and departure (D) of $R$ is $(0,0) \mathrm{km}$. What is the latitude and departure of $P$ (in km, rounded off to one decimal place)?
$\begin{array}{|c|c|c|c|c|} \hline Paths & RC & CA & AB & BP \\ \hline \begin{array}{c}\text {Directions} \\ \text {(in degrees)}\end{array} & 120 & 0 & 90 & 240 \\ \hline \end{array}$
 A $L=2.5 ; D=5.0$ B $L=0.0 ; D=5.0$ C $L=5.0 ; D=2.5$ D $L=0.0 ; D=0.0$
GATE CE 2023 SET-2   Geomatics Engineering
Question 1 Explanation:

Latitude of ' $C$ ' $=-5 \sin 30^{\circ}=-2.5 \mathrm{Km}$
Departure of ' $C$ ' $=5 \cos 30^{\circ}=+4.33 \mathrm{Km}$
Latitude of ' $\mathrm{B}$ ' $=$ Latitude of ' $\mathrm{C}$ ' $+5 \mathrm{Km}$ $=-2.5+5=+2.5 \mathrm{Km}$
Departure of 'B' $=$ Departure of ' $C$ ' $+5 \mathrm{Km}$ $=+4.33+5=+9.33 \mathrm{Km}$
Latitude of ' $P$ ' = Latitude of ' $B$ ' $-5 \sin 30^{\circ}$ $=+2.5-2.5=0$
Departure of ' $P$ ' $=$ Departure of ' $B$ ' $-5 \sin 30^{\circ}$ $=9.33-4.33=+5 \mathrm{Km}$
$\therefore \quad \mathrm{L}=0.0, \quad \mathrm{D}=5.0$
 Question 2
The direct and reversed zenith angles observed by a theodolite are $56^{\circ} 00^{\prime} 00^{\prime \prime}$ and $303^{\circ} 00^{\prime} 00^{\prime \prime}$, respectively. What is the vertical collimation correction?
 A $+1^{\circ}00'00''$ B $-1^{\circ}00'00''$ C $-0^{\circ}30'00''$ D $+0^{\circ}30'00''$
GATE CE 2023 SET-1   Geomatics Engineering
Question 2 Explanation:

In above example of both direct zenith angle and reversed zerwith angle are smaller than true value.

The error $(\varepsilon)=\left[\frac{\left(\phi_{1}+\phi_{2}\right)-360}{2}\right]$
$\therefore$ here, error $=\frac{\left(56^{\circ}+303^{\circ}\right)-360^{\circ}}{2}=-30^{\prime}$
$\therefore \quad$ Correction $=+30^{\prime}$

 Question 3
The diameter and height of a right circular cylinder are 3 cm and 4 cm, respectively. The absolute error in each of these two measurements is 0.2 cm. The absolute error in the computed volume (in $cm^3$, round off to three decimal places), is _______.
 A 1.65 B 7.52 C 3.25 D 5.18
GATE CE 2020 SET-2   Geomatics Engineering
Question 3 Explanation:
Let diameter, x = 3 and height = y = 4 and error= $\pm 0.2$
\begin{aligned} V&= \pi\left ( \frac{x}{2} \right )^2y=\frac{\pi x^2 y}{4}\\ V&=f(x,y) \\ dV&= \left ( \frac{\partial V}{\partial x} \right )dx +\left ( \frac{\partial V}{\partial y} \right )dy\\ dV&= \left ( \frac{1}{2}\pi xy \right )dx +\left ( \frac{\pi x^2}{4} \right )dy\\ &= \frac{1}{2} \pi \times 3 \times 4 \times (0.2)+\frac{\pi}{4} \times (3)^2 \times (0.2)\\ &= 1.65 \pi\\ &=1.65 \times 3.14=5.18 \end{aligned}
i.e., absolute error = |5.18| = 5.18
 Question 4
A series of perpendicular offsets taken from a curved boundary wall to a straight survey line at an interval of 6m are 1.22, 1.67, 2.04, 2.34, 2.14, 1.87, and 1.15 m. The area (in $m^2$, round off to 2 decimal places) bounded by the survey line, curved boundary wall, the first and the last offsets, determined using Simpson's rule, is _______
 A 35.5 B 45.25 C 68.5 D 75.45
GATE CE 2019 SET-2   Geomatics Engineering
Question 4 Explanation:

Area by Simpson's rule
\begin{aligned} A&=\frac{d}{3}[h_0+h_n+4(h_1+h_3+...)+\\ & 2(h_2+h_4+...)] \\ &= \frac{6}{3}[1.22+1.15+4 \times(1.67+2.34+1.87)+\\ & 2(2.04+2.14)]\\ &= 68.50m^2 \end{aligned}
 Question 5
Consider the hemi-spherical tank of radius 13 m as shown in the figure. What is the volume of water (in $m^3$) when the depth of water at the centre of the tank is 6 m?
 A $78 \pi$ B $156 \pi$ C $396 \pi$ D $468 \pi$
GATE CE 2019 SET-2   Geomatics Engineering
Question 5 Explanation:
\begin{aligned} \text{Volume of water}&=\frac{1}{3}\pi h^2(3r-h)\\ &=\frac{1}{3}\pi \times 6^2 \times (3 \times 13 -6)\\ &=396\pi \end{aligned}

There are 5 questions to complete.