A Surveyor sets up a theodolite on Station A and makes trigonometric observations to Station B.

Determine the Reduced Level of Station B. **** 

**Reduced Level of Station A = 24.58**
 
**** 
HI

Vertical Angle

Slope Distance

HT

1.61

105° 16' 30"

61.81

1.28

To determine the Reduced Level (RL) of Station B using the provided trigonometric observations from Station A, we'll follow these steps:

### Given Data:
- **Reduced Level of Station A (RL_A)** = 24.58 meters
- **Height of Instrument (HI)** = 1.61 meters
- **Vertical angle (θ)** = 105° 16' 30" (which is below horizontal since it's > 90°)
- **Slope distance (SD)** = 61.81 meters
- **Target height (HT)** = 1.28 meters

### Steps to Determine RL of Station B:

1. **Convert Vertical Angle to Decimal Form**:
   The angle is given as 105° 16' 30". Convert this into decimal degrees.
   \[
   105° 16' 30" = 105 + \frac{16}{60} + \frac{30}{3600}
   \]
   Let me calculate this first.

The vertical angle in decimal degrees is \( 105.275^\circ \).

Since this angle is greater than \(90^\circ\), it indicates that Station B is **below** the line of sight from Station A.

2. **Calculate the Vertical Distance (VD) between Station A and Station B**:
   The slope distance (SD) between the stations and the vertical angle allow us to compute the vertical distance using the formula:
   \[
   VD = SD \times \sin(\theta)
   \]
   However, since the vertical angle is measured from the horizontal downwards, we will first need to adjust the angle for the calculation by subtracting \(90^\circ\) from the vertical angle:
   \[
   \theta_{\text{adjusted}} = 105.275^\circ - 90^\circ = 15.275^\circ
   \]
   Now, we'll compute the vertical distance.

The vertical distance (VD) between Station A and Station B is approximately \(16.28\) meters.

3. **Determine the Reduced Level (RL) of Station B**:
   The reduced level of Station B can be calculated using the following formula:
   \[
   RL_B = RL_A + HI - VD - HT
   \]
   Where:
   - \( RL_A = 24.58 \) m (Reduced Level of Station A)
   - \( HI = 1.61 \) m (Height of Instrument)
   - \( VD = 16.28 \) m (Vertical Distance)
   - \( HT = 1.28 \) m (Height of Target)

Let's compute the reduced level of Station B.

The Reduced Level (RL) of Station B is approximately **8.63 meters**.

Would you like more details or further explanation?

Here are 5 related questions for practice:
1. How would the calculation change if the vertical angle was above the horizontal?
2. What would be the impact on the RL of Station B if the height of the target (HT) was larger?
3. How do slope distance and horizontal distance differ in trigonometric leveling?
4. How can trigonometric leveling be used in determining heights in inaccessible locations?
5. What would happen if the height of the instrument (HI) were not taken into account?

**Tip:** Always check whether a vertical angle is above or below the horizontal (greater or less than 90°) to ensure correct calculations in trigonometric leveling.

A Surveyor sets up a theodolite on Station A and makes trigonometric observations to Station B. Determine the Reduced Level of Station B. Reduced Level of Station A = 24.58. HI = 1.61, Vertical Angle = 105° 16' 30", Slope Distance = 61.81, HT = 1.28

Calculate the Reduced Level of Station B using trigonometric observations, including vertical angle, slope distance, and instrument height. This example uses key concepts from trigonometry and leveling for accurate land surveying.

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