Question 1188559: A surveyor starting from a point A moves N. 28°51′ E. a distance of 547′ 2.75″ to point B. Next, she moves N. 56°39′ E. a distance of 342′ 6.75″ to point C. Next, she walks S. 8°43′ W. a distance of 1567′ 6.25″ to point D. Finally, she returns to the starting point. What is the acreage of this plot of land?
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to calculate the acreage of the plot of land:
**1. Convert bearings to angles relative to the x-axis (East):**
* **AB:** N 28°51' E = 90° - 28°51' = 61°09' (Measured counter-clockwise from East)
* **BC:** N 56°39' E = 90° - 56°39' = 33°21' (Measured counter-clockwise from East)
* **CD:** S 8°43' W = 180° + 8°43' = 188°43' (Measured counter-clockwise from East)
**2. Convert distances to decimals of a foot:**
* AB: 547' 2.75" = 547.04583 ft (2.75/12)
* BC: 342' 6.75" = 342.5625 ft (6.75/12)
* CD: 1567' 6.25" = 1567.52083 ft (6.25/12)
**3. Calculate the coordinates of each point:**
We'll assume point A is at (0,0). We'll use the following formulas:
* Δx = distance * cos(angle)
* Δy = distance * sin(angle)
| Point | Distance (ft) | Angle | Δx (ft) | Δy (ft) | X (ft) | Y (ft) |
|---|---|---|---|---|---|---|
| A | 0 | 0 | 0 | 0 | 0 | 0 |
| B | 547.04583 | 61°09' | 266.25 | 480.71 | 266.25 | 480.71 |
| C | 342.5625 | 33°21' | 286.14 | 189.62 | 552.39 | 670.33 |
| D | 1567.52083 | 188°43' | -223.40 | -1549.48 | 328.99 | -879.15 |
**4. Calculate the area using the surveyor's formula:**
Area = 0.5 * |(x1y2 + x2y3 + x3y4 + x4y1) - (y1x2 + y2x3 + y3x4 + y4x1)|
Area = 0.5 * |(0 + 266.25\*670.33 + 552.39\*(-879.15) + 328.99\*0) - (0 + 480.71\*286.14 + 670.33\*(-223.40) + (-879.15)\*0)|
Area = 0.5 * | (178455.61 - 485686.69) - (137533.25-149870.32) |
Area = 0.5 * | -307231.08 - (-12337.07)|
Area = 0.5 * |-294894.01|
Area = 147447.005 sq ft
**5. Convert to acres:**
1 acre = 43,560 sq ft
Area = 147447.005 sq ft / 43560 sq ft/acre
Area ≈ 3.39 acres
**Answer:**
The acreage of the plot of land is approximately 3.39 acres.
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