Question

The height of the plane of collimation of a levelling instrument is 100.000 m from a datum. The levelling instrument measures a Back Sight of 1.500 m on a vertically held staff at a ground point \(P\). The reduced level (in m) of the ground point \(P\) with respect to the datum is (rounded off to three decimal places).

Hint:

Reduced level equals Height of Instrument minus the staff reading.

Answer 1

Correct Answer: 98.5

Step 1: Recall the Height of Collimation method in levelling.
In the Height of Collimation (or Height of Instrument) method, the reduced level of the line of sight of the levelling instrument, called the Height of Instrument (HI) or plane of collimation, is found first from a known benchmark. The reduced level (RL) of any staff point is then found by subtracting the staff reading taken at that point from the HI.
\[ RL_{point} = HI - \text{Staff reading} \]

Step 2: Identify the given quantities.
Here the plane of collimation is already given directly as \(HI = 100.000\) m above datum. The staff reading taken at point \(P\) is a Back Sight of \(1.500\) m. A Back Sight is simply the first staff reading taken from a given instrument position; it is subtracted from HI in exactly the same way as any other staff reading once HI is known.

Step 3: Apply the formula.
\[ RL_P = HI - BS = 100.000 - 1.500 \]
\[ RL_P = 98.500 \ \text{m} \]

Final Answer:
The reduced level of the ground point \(P\) with respect to the datum is 98.500 m.
\[ \boxed{RL_P = 98.500 \ \text{m}} \]