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).

The correct answer is 98.500.

In surveying, the plane of collimation (or height of instrument, HI) is the elevation of the line of sight of the telescope with respect to a reference datum. When a staff reading is taken on a point of unknown elevation, the reduced level (RL) of that point can be calculated using the relationship between the height of the plane of collimation and the staff reading.

The fundamental formula relating these parameters is:

$$\text{Height of Plane of Collimation (HPC)}=\text{Reduced Level (RL) of point P}+\text{Back Sight (BS)}$$
Rearranging the formula to solve for the Reduced Level (RL) of ground point P:

$$\text{Reduced Level (RL) of P}=\text{HPC}-\text{BS}$$

Given the values from the problem:
• Height of the plane of collimation (HPC) = 100.000 m
• Back Sight (BS) at point P = 1.500 m

Substituting these values into the rearranged equation:

$$\text{Reduced Level (RL) of P}=100.000-1.500$$
$$\text{Reduced Level (RL) of P}=98.500\text{m}$$

Thus, the reduced level of the ground point P with respect to the datum is 98.500 m.