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

Instead of quoting the HI formula directly, it helps to picture the actual vertical distances involved and build the answer from that picture.

The line of sight of the level (the plane of collimation) sits at a height of 100.000 m above the datum. This horizontal line of sight is a fixed, known elevation once the instrument has been set up and its height of collimation worked out, so it acts as a reference line for every staff reading taken from this instrument position.

At point $P$, the staff is held vertically and the instrument reads 1.500 m on it. This reading is the vertical distance measured downward from the horizontal line of sight (at 100.000 m) to the foot of the staff, which rests on the ground at $P$.

So the ground at $P$ must be lower than the line of sight by exactly this staff reading:

\[ \text{Elevation of } P = \text{Elevation of line of sight} - \text{staff reading} \] \[ RL_P = 100.000 - 1.500 = 98.500 \ \text{m} \]

The fact that this particular reading is called a Back Sight only describes when it was taken in the levelling sequence (the first reading from this instrument setup); it does not change how the reading is used. Any staff reading, back sight, intermediate sight or fore sight, is subtracted from the height of collimation in exactly this way to get the ground reduced level.

The reduced level of point $P$ is 98.500 m above the datum.

\[ \boxed{98.500 \ \text{m}} \]