 Land Surveying

Karyn Carson

According to Wikipedia, land surveying is “the technique and science of accurately determining the terrestrial or three-dimensional space position of points and the distances and angles between them.” There are several types of surveys: boundary, topographical, as-built, construction, and geological are just a few.  Mathematics is used throughout surveying and is instrumental in determining the position of objects.  Also, mathematics is used in the business side of surveying.  Many people who are surveyors have their own business and must know how their company is run.  How much do they spend on labor, benefits, facilities?  How much should they charge for a survey?

I’ve looked at Principles and Practice of Land Surveying:  Sample Examination by George M. Cole (Professional Publication, Inc., 2003) and collected several problems to show some of the ways mathematics is used in surveying.

1.       Land Description

Lot 1, Block A of Silver Lake Plantation Subdivision as recorded in Plat Book 1 of the public records of Jefferson County, Florida;

Less and Except the western 1/4 of said Lot 1; and

Less and Except a portion of said Lot 1 described as follows:  Begin at the intersection of the south line of said Lot 1 and the west line of Section 12, Township 2 North, Range 5 East; then go northerly along said section line for 250 ft; then go easterly in a straight line for 660 ft more or less to the east line of said Lot 1; then go southerly for 250 ft along said east line to the southeast corner of said Lot 1; then go westerly along the south line of said Lot 1 for 500 ft more or less to the Point of Beginning; containing 3.79 ac more or less.

Lot 1 Dimensions per Survey a.   What is the total acreage of Lot 1 based on the survey data?

b.   What is the acreage of the land excepted from Lot 1 by the first Less and Except parcel? Solutions:

a.   To solve this problem, we would first have to separate the piece of land into three shapes:  a rectangle and two right triangles.

 1320 ft By creating perpendicular segments to the northern side of the property,  we create three segments which now become the bases of our rectangle and two right triangles.

The base of the rectangle is 1320 feet (660.0 + 660.0) and the height is 660 feet; therefore the area of the rectangle is 871200 sq. ft.  Now, let’s look at the triangles.

We must find the measurements of the bases of the two triangles.  How can this be determined?

To find the base of the western triangle, simply subtract 660 ft from 671.52:

671.52-660 = 11.52 ft

To find the base of the eastern triangle, subtract 660 from 683.05:

683.05-660 = 23.05 ft

Remember, to find the area of a triangle, you take half the product of the base and the height.

Area of western triangle: Area of eastern triangle: And now to find the total area:

Atotal = 871200 + 3801.6 + 7606.5

Atotal = 882608.1 sq. ft

Because there are 43560 sq. ft in an acre, we must divide the total area by this number to calculate the number of acres in our property: b.   To find the Less and Except portion of the lot, just find ¼ of the total acreage: Another problem from the Sample Examination:

2.   Land Description

All land in Section 7 lying northerly and easterly of State Road 99: What is most nearly the distance along the centerline of State Road 99 within section 7?

Solution:

For curve 1, find the arc length, L, by taking the measure of the interior angle, dividing it by 360 degrees and multiplying by 2(pi)r:  For curve 2, do the same: The total distance, Dtotal, is:

Dtotal = tangent1 + L1 + L2 + tangent2

= 500 ft + 1745.33 ft + 1309.00 ft + 1500 ft

= 5054.33 ft

Also, from a business sense, there are many instances where knowledge of mathematics will come in handy.

Results from an audit for a firm are as follows:

 Item Cost (\$) Direct labor 3,682,957 Allowable fringe benefits 1,577,283 Allowable general overhead 4,684,247 Facility cost of capital 90,621

1.    What is most nearly the factor for combined fringe benefits and general overhead for the firm on a direct labor basis?

The total overhead factor is the ratio of the sum of fringe benefit costs and general overhead costs to direct labor costs: This is equal to 170%.

2. For a project involving \$250,000 in direct labor costs, what would most nearly be the total cost to the firm of producing the services for the project neglecting facility cost of capital?

To find the project cost, you must first find the product of the direct labor cost and the overhead rate, then add that to the direct labor cost: 3. Assuming that a 15% operating margin on total cost is desired, approximately what fee should be charged by the firm for the project described in the previous problem, neglecting facility cost of capital?

Well, 15% of \$675,000 is \$101,250 – now add that amount to the project cost:

\$675,000 + \$101,250 = \$776,250

4. What is most nearly the facility capital cost of money (FCCM) rate for the firm, on a direct labor basis?

We need to compare the facility cost to the direct labor: Approximately 2.5% of the direct labor for the project should be charged for the facility cost.

5. How much should this be for the project outlined in the previous problems?

2.5% of \$250,000 is \$6,250.

6. What is the total bid for this project?

\$675,000 + \$101,250 + \$6150 = \$782,400