To determine the first 6 offsets to be set out at 10mm intervals along a tangent to locate a curve of radius 400m, we can use two methods: the radial method and the perpendicular offset method. Both methods involve calculating the distances from the tangent to the curve at regular intervals.

a) Radial Method:
In the radial method, the offsets are measured along lines radiating from the center of the curve. To calculate the offsets using this method, the following formula can be used:

Offset distance = Radius of curve × Tangent of the angle (θ)

Where θ is the angle between the tangent and the radial line. In this case, we want to set out the offsets at 10mm intervals, so we need to determine the angles at which these intervals occur.

Step 1: Calculate the angle at the first offset:
Angle (θ) = Offset distance / Radius of curve = 10mm / 400m = 0.025 radians

Step 2: Calculate the angle for subsequent offsets:
Angle (θ) = (10mm × Offset number) / Radius of curve

Using this formula, we can calculate the angle for each offset. The first 6 offsets will have the following angles:

Offset 1: θ = 0.025 radians
Offset 2: θ = (10mm × 2) / 400m = 0.05 radians
Offset 3: θ = (10mm × 3) / 400m = 0.075 radians
Offset 4: θ = (10mm × 4) / 400m = 0.1 radians
Offset 5: θ = (10mm × 5) / 400m = 0.125 radians
Offset 6: θ = (10mm × 6) / 400m = 0.15 radians

b) Perpendicular Offset Method:
In the perpendicular offset method, the offsets are measured perpendicular to the tangent line. This method involves calculating the distances between the tangent and the curve at regular intervals.

To determine the offsets using this method, we can use the following formula:

Offset distance = Radius of curve - √(Radius of curve^2 - Distance along tangent^2)

Step 1: Calculate the distance along the tangent for the first offset:
Distance along tangent = Offset distance = 10mm

Step 2: Calculate the distance along the tangent for subsequent offsets:
Distance along tangent = (10mm × Offset number)

Using this formula, we can calculate the distance along the tangent for each offset. The first 6 offsets will have the following distances along the tangent:

Offset 1: Distance along tangent = 10mm
Offset 2: Distance along tangent = 10mm × 2 = 20mm
Offset 3: Distance along tangent = 10mm × 3 = 30mm
Offset 4: Distance along tangent = 10mm × 4 = 40mm
Offset 5: Distance along tangent = 10mm × 5 = 50mm
Offset 6: Distance along tangent = 10mm × 6 = 60mm

Both methods provide a way to set out offsets at regular intervals along a tangent to locate a curve of radius 400m. The choice between the radial method and the perpendicular offset method depends on the specific requirements and constraints of the project.