Bearing and directions

Question

Bearings: (Start on the right of the paper) Beginning at the first survey marker (A), proceed 227.0 feet bearing due South to marker B. At marker B, turn down West and proceed 127.0 feet to marker C. At marker C, turn to a bearing N13.2°W and proceed 151.2 feet to marker D. At marker D, turn on a bearing of N42°E and proceed 122.6 feet to marker E. At marker E, turn to bearing S81.9°E and proceed 80.3 feet, which returns to the point of beginning.

 

Now, for the actual task.

 

Task: Make a sketch of the property (it need not be a scale drawing). Calculate the exact area of the tract. You have discovered that in order to develop the property, the company will have to run a gas line from marker D to marker A, and a water line from Marker C to Marker A. Report the exact lengths of the gas and water lines. Excavating for the lines will cost $23/foot (same cost for gas or water). Installing the water will cost $.35 per foot, and installing the gas line will cost $1.20 per foot. Estimate the total cost for installing the gas and water lines. 

 

Hints: Begin drawing near the upper right corner of your paper. To find the area, split the irregular pentagon up into three triangles. Use the laws of sines and cosines to find the lengths of the sides you construct. Find the area of the triangle using the formula

A = 1/2 bc sin C

I have everything solved but I was unsure if the middle triangle area equals 13268.5 ft²

I hope someone can help me confirm if this information is correct which will end me with a total area of 31768.2 ft²

Answer 1

This one gets much easier if you stop thinking in bearings and start thinking in coordinates. Put marker A at the origin, let the positive x direction be East and the positive y direction be North, and then walk the boundary one leg at a time, adding the east and north pieces of each leg as you go.

For a bearing written like NθW, the leg moves cosθ of its length toward the north and sinθ of its length toward the west, and the same idea handles the other three quadrants. Walking the legs in order gives you:

A = (0, 0)

B = (0, −227.0), since due south is straight down

C = (−127.0, −227.0), since due west is straight left

D = (−127.0 − 151.2 sin13.2°, −227.0 + 151.2 cos13.2°) = (−161.53, −79.79)

E = (−161.53 + 122.6 sin42°, −79.79 + 122.6 cos42°) = (−79.49, 11.31)

Before going any further, use the last leg as a check. From E, a bearing of S81.9°E for 80.3 feet moves you 80.3 sin81.9° = 79.49 to the east and 80.3 cos81.9° = 11.31 to the south, which lands you at (0.01, 0.00). That is marker A again, so your traverse closes and you know every coordinate above is right. Surveyors do exactly this check, and it is worth doing before you calculate anything.

The sketch is now easy, because you just plot those five points and join them in order. You get a five sided lot with a long south edge, a long west edge running up and slightly left, and two shorter edges cutting back east to A.

For the area, use the shoelace formula on the coordinates you just built, going around A, B, C, D, E in order:

Area = ½ |(x_Ay_B − x_By_A) + (x_By_C − x_Cy_B) + (x_Cy_D − x_Dy_C) + (x_Dy_E − x_Ey_D) + (x_Ey_A − x_Ay_E)|

That comes out to about 31,766 square feet, which is roughly 0.73 acres.

The two line lengths are just the distance formula, and they are especially clean here because A sits at the origin, so the distance is simply the square root of the sum of the squares of the other marker's coordinates.

Gas line, D to A: √(161.53² + 79.79²) ≈ 180.2 feet

Water line, C to A: √(127.0² + 227.0²) ≈ 260.1 feet

Together that is about 440.3 feet of trench, so at $23 per foot the excavation runs about $10,126. Your question got cut off where the installation costs start, so once you have those numbers you would add them on top of that figure.

So, the method to remember is this. Turn every bearing into an east piece and a north piece, add them up to get coordinates, confirm the traverse closes back at your starting marker, and after that the area is just the shoelace formula and the lengths are just the distance formula.