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To find the included angle ∠BAC\angle BAC∠BAC between two lines AB and AC, given their whole circle bearings, you subtract the smaller bearing from the larger one:

Given:

• Whole circle bearing of line AB: 115∘115^\circ115∘
• Whole circle bearing of line AC: 41∘41^\circ41∘

The included angle ∠BAC\angle BAC∠BAC is calculated as:

∠BAC=115∘−41∘=74∘\angle BAC = 115^\circ - 41^\circ = 74^\circ∠BAC=115∘−41∘=74∘

So, the included angle ∠BAC\angle BAC∠BAC is:

74°

At the equator, the magnetic dip (or inclination) is:

0°

The magnetic dip is 0° at the equator because the Earth's magnetic field is horizontal to the surface at this location.

The correction to be applied to each 30 m chain for a line measured along a slope of θ\thetaθ is:

30 (1 - cos θ\thetaθ)

This formula accounts for the difference between the actual measured length along the slope and the horizontal distance.

Contour lines cross ridge or valley lines at:

90°

This perpendicular intersection ensures that the contour lines accurately represent the elevation changes and the shape of the ridge or valley.

The method of leveling in which the heights of mountains are found by observing the temperature at which water boils is known as:

Hypometry

Hypometry involves measuring the elevation of a location by determining the boiling point of water, which changes with atmospheric pressure and altitude.

The rise and fall method is used in:

Differential levelling

This method is employed to determine the differences in elevation between various points by calculating the rise and fall of the instrument's line of sight between observations.

According to Lehman's rule of plane tabling:

All of the above

Lehman's rule states that:

1. When looking in the direction of the reference points, the point sought is on the same side of each of the three rays.
2. The distance to the point sought from each of the three rays is proportional to the distance of the three known points from the instrument station.
3. When the instrument station is outside the great circle, the point sought is always on the same side of the ray drawn to the most distant point as the intersection of the other two rays.

When the latitudes and departures are adjusted so that the algebraic sum of the latitudes and departures is equal to zero, the operation is called:

Balancing the traverse

Balancing the traverse involves adjusting the latitudes and departures to ensure that the survey is closed and the sums are balanced, which helps in minimizing errors and ensuring accurate results.

The deflection angle may have any value between:

0° and 180°

The deflection angle measures the angle between the direction of the previous line and the direction of the new line, and it can range from 0° to 180°, depending on the direction of the turn.

The distance between the midpoint of the long chord and the midpoint of the curve is equal to:

R (1 - cos(θ/2))

Here, RRR is the radius of the curve and θθθ is the central angle of the curve. This formula calculates the distance from the midpoint of the chord to the midpoint of the curve (i.e., the curve's arc) based on the radius and the angle subtended by the chord.