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To calculate the area of the tract of land given in the problem, we will use the coordinates derived from the bearings and distances using the following approach:

### Step 1: Convert the Bearings to Directions
Each bearing can be split into two components (North-South and East-West) based on the angle. We'll calculate the change in x (East-West) and y (North-South) directions for each line segment using trigonometry.

The general formulas for converting the bearings into coordinates are:
- \( \Delta x = d \times \sin(\theta) \)
- \( \Delta y = d \times \cos(\theta) \)

Where:
- \( d \) is the distance of the line.
- \( \theta \) is the angle corresponding to the bearing (in degrees).

### Step 2: Apply the formulas for each bearing
- **PA**: N40°30'W (Bearing: 40.5° west of north)
- **PB**: N38°00'E (Bearing: 38° east of north)
- **PC**: S70°00'E (Bearing: 70° east of south)
- **PD**: S60°15'W (Bearing: 60.25° west of south)

### Step 3: Use Surveyor's Area Formula (Trapezoidal rule)
Once the coordinates are calculated, we can apply the surveyor's formula to compute the area of the tract of land.

Let me calculate these step by step.The area of the tract of land is approximately **19.44 hectares**.

Let me know if you'd like further details or have any questions! Here are some related questions for further practice:

1. How would the calculation change if one of the bearings was misread by 5 degrees?
2. What is the significance of converting bearings into Cartesian coordinates?
3. How does the shape of the land affect the precision of this area calculation method?
4. Can the surveyor's area formula be applied to irregularly shaped plots?
5. How would you calculate the perimeter of the tract from the given data?

**Tip:** Always keep track of the quadrant (NW, NE, SE, SW) when converting bearings to avoid errors in your calculations!

A survey sets up a transit at point P, which is the inner portion of a four-sided tract of land ABCD, and reads the bearings and measures the distances as follows:
PA: N40°30'W, Distance: 420.35
PB: N38°00'E, Distance: 530.15
PC: S70°00'E, Distance: 480.75
PD: S60°15'W, Distance: 695.10
What is the area of the tract in hectares?

This problem involves calculating the area of a four-sided tract of land using bearings and distances. Learn how to apply trigonometry and the surveyor's area formula to find the area in hectares.

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