---
title: "Gradient between X and Y from R"
subject: "Surveying II"
chapter: 2
key: "semester_4/surveying_ii:tacheometer-gradient-xy-from-r-2071b"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/tacheometer-gradient-xy-from-r-2071b
site: BCT Engineering
---

# Gradient between X and Y from R

Surveying II · Chapter 2: Tacheometry

- Asked in: 2071 Bhadra, 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/tacheometer-gradient-xy-from-r-2071b
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

A tachometric survey was done to find the gradient between X and Y. Tacheometer consist of an anallatic lens was used and following observations were made from section R on vertical staff.

| Inst. Stn. | Staff point | Stadia hair readings | Vertical angle | Bearing |
|---|---|---|---|---|
| R | X | 0.915, 1.750, 2.585 | +15° | 345° |
| | Y | 0.760, 2.240, 3.715 | +10° | 75° |

## Short answer

Using D=Ks cos²θ and V=½Ks sin2θ, then cosine rule for XY. XY = 326.21 m, ΔH = +8.293 m, gradient rising 1 in 39.3 (2.54%).

## Answer

Anallatic lens: $C=0$, $K=100$. The middle hair reading is the axial reading $h$. The height of instrument is not given, so only the difference in level between X and Y is found (it cancels).

Staff held vertical, line of sight inclined at angle $\theta$ (additive constant zero, which is the case for an anallatic lens or $C=0$):

$$
D=Ks\cos^2\theta,\qquad V=\tfrac12Ks\sin2\theta
$$

where $s$ is the staff intercept, $\theta$ the vertical angle and $K=100$, $C=0$.

#### Step 1: Horizontal distance and vertical component of each sight
**Point X** (bearing 345°00', angle of elevation $15^\circ 00'$):

$$
\begin{aligned}
s&=2.585-0.915=1.670\ \text{m}\\
D&=100\times1.670\times\cos^2 15^\circ 00'=155.813\ \text{m}\\
V&=\tfrac12\times100\times1.670\times\sin 2(15^\circ 00')=41.750\ \text{m}
\end{aligned}
$$

**Point Y** (bearing 75°00', angle of elevation $10^\circ 00'$):

$$
\begin{aligned}
s&=3.715-0.760=2.955\ \text{m}\\
D&=100\times2.955\times\cos^2 10^\circ 00'=286.590\ \text{m}\\
V&=\tfrac12\times100\times2.955\times\sin 2(10^\circ 00')=50.533\ \text{m}
\end{aligned}
$$

#### Step 2: Difference in level
The height of instrument is the same for both sights, so it cancels (RL of point $=$ RL of axis $+V-h$):

$$
\begin{aligned}
\Delta H&=(V_{Y}-h_{Y})-(V_{X}-h_{X})\\
&=(+50.533-2.240)-(+41.750-1.750)=+8.293\ \text{m}
\end{aligned}
$$

#### Step 3: Horizontal distance XY
The angle at R between the two lines is the difference of the bearings, $90^\circ 00'$ (345°00' and 75°00').

$$
\begin{aligned}
XY&=\sqrt{D_1^2+D_2^2-2D_1D_2\cos\gamma}\\
&=\sqrt{155.813^2+286.590^2-2\times155.813\times286.590\cos 90^\circ 00'}=326.207\ \text{m}
\end{aligned}
$$

#### Step 4: Gradient
$$
\text{Gradient}=\frac{\Delta H}{XY}=\frac{8.293}{326.207}=0.02542
$$

The line from X to Y is rising: 1 in 39.3, i.e. 2.54 %, an angle of $1^\circ 27'23''$ with the horizontal.

**Answer:** XY = 326.21 m; difference in level = +8.293 m; gradient rising 1 in 39.3 (2.54 %, 1°27').

## Check yourself

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

1. What are K and C for this instrument?
   A. K = 100 and C = 0 because the lens is anallatic
   B. K = 100 and C = 0.3
   C. K = 0 and C = 100
   D. K = 1 and C = 0
   Answer: A. An anallatic lens eliminates the additive constant. (Key point: Constants)

2. What is the intercept at X (0.915, 1.750, 2.585)?
   A. 1.750 m
   B. 1.670 m (top - bottom)
   C. 0.835 m
   D. 3.500 m
   Answer: B. s = 2.585 - 0.915 = 1.670 m. (Key point: Staff intercept)

3. What is the angle XRY used in the cosine rule?
   A. 270°
   B. 60°
   C. 90° (difference of bearings 345° and 75°)
   D. 75°
   Answer: C. The difference between 345° and 75° is 270°, whose smaller included angle is 90°. (Key point: Angle between lines)

4. How is the level difference of Y over X computed?
   A. VY - VX only
   B. (VY + hY) - (VX + hX)
   C. hY - hX
   D. (VY - hY) - (VX - hX) = +8.293 m
   Answer: D. Each RL = axis + V - h, and axis RL cancels; result +8.293 m, so XY is rising. (Key point: Level difference)

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