Find the number of acres of a piece of land described as follows:

Question

From the iron post at the old mill, proceed 85 rods south, then 67 rods in the direction N80E, then 120 rods N10W to the Mill road, then along Mill road to the starting point.

 

1 rod=16.5 ft, 1 acre=43,560 sq.ft.

Answer 1

What you have here is a quadrilateral (four-sided polygon) ABCD with the following information (A is the vertex representing the mill):

 

The leftmost edge AB runs north-south and has length 85 rods = 1402.5 ft.

The bottom edge BC has length 67 rods = 1105.5 ft and makes an angle ∠ABC = 80°.

The rightmost edge CD has length 120 rods = 1980 ft and makes an angle of 10° with the north-south line with D further west than this line (C is on the line).

The top edge AD has unknown length (at present).

 

To find the acreage, you must find the area of this figure, then convert it to proper units. The best way to do this is to add the areas of the two triangles ABC and ACD. This will require some additional problem solving.

 

In ABC, you know SAS, so you can use the formula K = (1/2) (AB)(BC) sin 80° = 2804.24 ft².

 

You also need the length AC, which can be found with the Law of Cosines. The answer is AC = 98.672 rods = 1628.1 ft.

And, the angle BCA can be found using the Law of Sines. Its measure is 58.0°.

 

The angle ACD is 180° - 80° - 58.0° - 10° = 32°.

 

Now, you can find the area of ACD using the formula K = (1/2) (AC)(CD) sin 32° = 3137.29 ft².

 

The sum of the two areas is ABCD = 2804.24 + 3137.29 = 5941.53 ft² = 5941.53/43560 = .1364 acres.

 

After all this, I made have made a mistake. Also, you did not specify significant digits, so I would give the final answer as .14 acres since the given information is accurate to 2 significant digits.