Geometry
Animal Systems
Lesson Preview
This lesson focuses on area, sides and angles of right triangles, and calculations of irregular shaped areas. These concepts will be placed in the agricultural context of overgrazing and stocking rates, grazing techniques used with differing species, and common methods of land measurement.
TEKS Alignment
130.10.c.3C use geometric principles to solve problems inherent to animal systems such as square footage for housing requirements; acreage calculation for normal and irregular shaped pastures; the use of right triangles for perpendicular cross fencing; calculation of feed bin volume based upon shape such as cylinder, cone, cube, or pyramid; and housing volume calculations for ventilation.
Lesson Objectives
Learners...
Introduction
When raising animals that need grazing pastures it is important to manage the animal to pasture ratio correctly. A livestock production manager must balance the number of animals with the current availability of forage. Management techniques such as grazing patterns and stocking rates play a vital part in finding the correct balance.
When determining grazing patterns for pastures, agriculturists often times have to calculate areas of irregular shapes. To do this, the Pythagorean Theorem is used. Livestock managers can also use surveying methods to measure land areas, slope, or runoff areas. This, too, requires the use of geometry. Other geometry applications are also used when constructing barns and other facilities for livestock use.
The pictures below show rotational grazing techniques being used in a beef cattle operation.
Content
Objective 1- Explain overgrazing and stocking rates
Overgrazing is caused by having too many animals in a particular area, or by not properly controlling their grazing activity. Overgrazing reduces plant leaf areas, which reduces interception of sunlight and plant growth. Plants become weakened and have reduced root length, and the pasture sod weakens. The reduced root length makes the plants more susceptible to death during dry weather. The weakened sod allows weed seeds to germinate and grow.
When there are too many animals in a pasture, soil may be visible between plants, and erosion is likely to occur. Reduced soil depth, soil organic matter, and soil fertility hurt future land efficiency. The effects of erosion, such as loss of soil depth and organic matter can take several years to correct. Erosion also influences the soil's water-holding capacity, and determines plant survival rates during dry conditions.
Overgrazing will in turn affect livestock performance and condition. Cows that have inadequate pasture grazing will not have a chance to gain weight after their calves are weaned, and may have poor body conditions. Also, cows in poor body condition do not cycle as soon after calving, which can result in delayed breeding. Poor weaning weights in calves can also be caused by overgrazed pastures.
The term stocking rate refers to the number of animals your land can support. In order to find the stocking rate two factors must be determined; the amount of forage needed for the consumption of the particular species, and how much available forage you have.
For more information on stocking rate formulas and examples of stocking rate formulas visit the following web site.
Complete the following crossword puzzle to review overgrazing and stocking rate information.
Watch this video for more information on grazing.
Objective 2- Summarize various rotational grazing techniques used with differing species;
Grazing management is the human manipulation of grazing livestock and the pasture resource to produce a profit for the manager and still maintain the productivity of the animals, pasture, and long-term stability of the land.
The most common grazing systems used for various livestock include:
Continuous Grazing- A group of animals has continual access to an area of land over a period of time. If the number of animals remains the same, this method is called set stocking. Continuous stocking is probably the most common grazing method.This type of pasture grazing requires the least amount of investment in time and money because of its ease of operation. The graphic below shows an example of continious grazing.
Rotational Grazing- Or rotational stocking, is a method that uses frequent periods of grazing and rest among two or more pastures for some set time period. A common rotational grazing system has two to four pastures, with animals grazing a pasture for about seven days or less and then being moved to the next pasture. This allows a rest period from ten to thirty days during each cycle. With rotational grazing management, pasture plants benefit from rest with more growth and vigor, animals gain from a more stable and more nutritious forage supply, and manure is spread more uniformly. Rotational grazing requires more fences and slightly more labor than continuous grazing. The graphic below shows an example of rotational grazing.
Management Intensive Grazing- A rotational grazing system gaining interest and acceptance is one that relies on pastures being divided into numerous paddocks. This type of grazing system has many names. The name gaining favor with producers and grazing advisors is management intensive grazing, or MIG. A key difference between rotational grazing through 2 or more pastures and management intensive systems is that the latter emphasizes more management of forage consumption, quality, and re-growth. The graphic below shows an example of management intensive grazing.
Strip Grazing- Useful to help prevent bloat in pastures containing a high proportion of legumes because it forces animals to eat both the leaves and the stems. Producers want their livestock to have fresh, leafy, high quality forage at all times will use temporary fence to ration only a day's or even half day's forage supply in strip razing system that may be the equivalent of a 50 or 60 pasture rotation.The graphic below shows an example of management strip grazing.
Variable stocking- Using any of the above systems, where a few animals are added or withdrawn from the main group to better match the animal use to the varying forage supply.
Leader-follower grazing- Or first-last grazing, where the first group of animals (with the highest nutritional needs) has first access to the best forage in each new pasture, followed by a second group of animals (whose nutritional needs are lower) to graze the less desirable forage remaining in the pasture. In the leader-follower method, animals can be different groups of the same species or groups of different species.
Take the following quiz over various rotational grazing techniques.
This video contains more information on rotational grazing.
Objective 3- Describe common methods of land measurement used in agriculture;
In the agricultural setting land is measured in acres. There are 43,560 square feet in an acre.
The terms sections and quarters are also sometimes referred to when measuring land. A section is 640 acres and a quarter is 160 acres. Therefore there are 4 quarters in a section.
Leveling and Surveying:
The determination of elevations is called leveling
A level consists of a high-powered telescope that is attached to a spirit or bubble level that keeps the line of sight of the telescope horizontal.
Basic surveying terminology:
Calculations of Elevations:
When surveying it's important to remember two basic geometry principles
1) two points are required to form a line
2) the shortest distance between two points is a straight line.
Visit this very useful website to learn the history, and how-to of surveying land.
Methods of land measurement:
Pacing- determine the distance by counting your steps or paces. A pace is defined by one step, while a stride is known as two steps.
Advantages- inexpensive, Fast, and Fairly accurate
Disadvantage- Pacing up or down a slope may cause greater inaccuracy,impossible terrain may cause problems, loose soil may distort results, and temperature, shoes, etc. may change results.
Chaining- A long metal chain that provides the distance on the chain links. A surveyor's chain is sixty-six feet in length, and is composed of one hundred links. The links are concocted by two rings, with a tally mark at the end of every ten links. A link in measurement includes a ring at each end, and is 7.95 inches long. Only linear measurements are taken, no angular measurements are taken. In chaining, eleven marking pins are needed, made either of iron, steel or brass wire, as preferred. They are about fourteen inches long, pointed at one end to enter the ground, and formed into a ring at the other end for convenience in handling.The chaining method would be desirable in areas that are open with fairly level ground. Large, wooden, and crowded areas would be less desirable conditions for chaining, and other methods should be considered for land measurement.
Chaining facts:
80 chains = 1 mile
10 square chains = 1 acre
4 rods = 1 chain
Taping- Fiberglass or cloth tapes used for less precise measurements.
Cloth- tend to stretch somewhat when wet. Some cloth have copper strand reinforcement
Fiberglass – rugged and consistent length
Steel taping- Heavy duty and most often used for route surveys. 100' is the most common.
Measuring wheel and odometer- The surveyor's wheel measures the distance along a surface, whereas in normal land surveying, distances between points are usually measured horizontally with vertical measurements indicated in differences in elevation. The measuring wheel has a mechanical or electronic counter to calculate the distance. The mechanical counters offer only a single unit of measure, where as the electronic counters offer multiple units of measure all in the same model of measuring wheel. The surveyor's wheel is not the most accurate form of land measurement, but it is used because it's fast and only needs one person to operate.
Stadia level- Uses telescope cross-wire to determine horizontal distance. Stadia is useful in surveying inaccessible terrain and in checking more precise measurements
Range Finders- Optical handheld device that uses a series of lenses and mirrors to produce a double image. The double images on mirrors are brought together by rotating a dial until both images merge into one. The dial has a distance indicator. When the two images merge, the distance to the target is read directly off the dial.
Electric Range finders or EDM- Can be handheld or tripod mounted devices that are electro-optical, and use a laser or infrared light source. The light beam is separated into wavelengths of a certain frequency. Measurement of distance is found by setting up the transmitter at one end of the line to be measured and a reflector at the opposite end. The time taken for the light wave to be sent to the target and reflect back gives the distance. Electric range finders are useful for areas that are difficult to measure or long in distance.
Look through the photo album to see pictures of land measurement tools and devices.
Watch this video to learn about measuring distance.
Complete the following activity to review land measurement terms and tools.
Objective 4: Calculate the area, sides, and angles of right triangles
The Pythagorean Theorem is all about squares. Simply stated, in a right triangle, the square built with side length of the hypotenuse has the same area as the sum of the areas of the squares built on the legs of the triangle (the two sides that form the right angle).
This site provides excellent animation describing the many proofs of the Pythagorean Theorem.
In the diagram, triangle ABC is a right triangle with right angle at C. Segment AC has length b (area of the square with side length b is b 2), segment BC has length a, and segment AB (the hypotenuse) has length c. Writing the Pythagorean Theorem as a mathematical sentence:
c 2 = a 2 + b 2
If any two sides of a right triangle are known, the third side can be found.
Example 1: The two legs of a right triangle measure 12 feet and 15 feet. What is the length of the hypotenuse to the nearest 10th of a foot?
c2 = 122 + 152
c2 = 144 + 225
c2 = 369
c ≈ 19.2 feet (take the square root of both sides)
The hypotenuse is 19.2 feet.
Example 2: How high up the wall will a 20-foot ladder touch if the foot of the ladder is placed 5 feet from the wall?
The ladder is the hypotenuse of a right triangle with the distance from the wall one leg.
20 2 = 5 2 + b 2
Rearranging gives:
b2 = 202 - 52
b2 = 400- 25
b2 = 375
b ≈ 19.4 (take the square root of both sides)
The top of the ladder will touch the wall about 19.4 feet up the wall.
Example 3: A rectangular garden 6 yards wide has a diagonal measuring 10 yards. Find the perimeter of the garden in feet. Find the area of the garden in square feet.
First find the length of the garden using the Pythagorean Theorem.
10 2 = 62 + b 2
b2 = 102 - 62
b2 = 100- 36
b2 = 64
b = 8 (take the square root of both sides)
The length of the garden is 8 yards.
To find the perimeter use
P = 2l + 2w
P = 2*8 + 2*6
P = 16 + 12
P = 28 yards
Converting to feet, there are 3 feet in each yard, so
P = 28*3
P = 84 feet
The perimeter of the garden is 84 feet.
To find the area use
A = l*w
A = 8*6
A = 48 square yards
Converting to square feet, there are 9 square feet in each square yard, so
A = 48*9
A = 432 square feet
The area of the garden is 432 square feet.
Note: Many will multiply square yards by 3 in an attempt to get square feet, but one must remember that the yards measurements are in two different directions
Objective 5: Calculate irregular shaped areas;
Area measure is used to measure surfaces. Units can be square inches, square feet, square yards, square miles, or acres. Students tend to forget area refers to a covering because of the relationship of linear measure used in the formulas to get the number of square units. It should be emphasized that a linear measure in two directions will give square units.
Some of the areas often measured include areas of regular and non-regular polygons and circles.
Below is a compilation of the basic area formulas often used.
Formulas:
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Formula |
Notes |
Picture |
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Area of a rectangle A = length * width
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Tic marks on a diagram indicate congruent (same length) sides. Sides of a rectangle are perpendicular. A rectangle is a special case of the parallelogram. |
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Area of a square is a special case of the rectangle A = side squared
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All sides of a square have the same length. |
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Area of a triangle
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Height refers to the perpendicular distance from the opposite vertex. Any side can be the base, and the height/altitude is the distance from that base to the opposite vertex. Sometimes the base has to be extended because the perpendicular falls outside the triangle.
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Area of a parallelogram A = base * height |
Height refers to the perpendicular distance between the two sides used as the base.
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Area of a trapezoid
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Height is the perpendicular distance between the two parallel bases. |
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Area of a circle A = π * radius squared
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Example 1: Find the area of the irregular field in square feet and acres.
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This irregular shaped field can be decomposed into a square and a right triangle.
A = 540*540 + ½ (540 + 360) * 360 Note: 540 * 540 is the area of the square region
540 + 360 is the "height" of the triangular region
½ (540 + 360) * 360 is the area of the triangular region
A = 291600 sq. feet + 162000 sq. feet
A = 453600 square feet
To change to acres, multiply square feet by 0.00002296.
A = 453600 * 0.00002296
A = 10.4147 acres
Example 2: The region below can be decomposed into a rectangle and a semicircle. Find the area in square feet and acres.
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A = 750 * 1000 + ½ * π * 3752 750 * 1000 gives the area of the rectangle
Circle has a radius of ½ * 750 = 375
Area of ½ circle is ½ π * r2
A = 750000 + 220893
A = 970893 square feet
Converting to acres:
A =970893* 0.00002296
A = 22.29 acres
Problem Solving Activity
An irregularly shaped pasture is pictured below.
1. What is the perimeter of the pasture to the nearest foot?
2. What is the area of the pasture in square feet to the nearest square foot?
3. What is the area of the pasture in acres to the nearest acre?
4. If two acres of pasture will support one cow, how many cows can be pastured?
Answers
1. What is the perimeter of the pasture?
Approximately 7055 feet
2. What is the area of the pasture in square feet?
Approximately 2,139,943
3. What is the area of the pasture in acres?
Approximately 49 acres
4. If two acres of pasture will support one cow, how many cows can be pastured?
Pasture will support 24 cows.
Best Practices
Additional Resources
http://www.clarkcd.org/pdf/Pasture%20Systems%20and%20Grazing%20Methods2.pdf
Introduction picture 1- http://msucares.com/news/print/agnews/an04/040226.html
Introduction picture 2- http://www.tjswcd.org/Agriculture.html
Rotational grazing picture- http://www.esc.rutgers.edu/publications/stablemgt/fs368.htm
http://www.takayaiwamoto.com/Pythagorean_Theorem/Pythagorean_Theorem.html
http://www.landsurveyors.com/
http://extension.usu.edu/
http://www.wvu.edu/~agexten/forglvst/overgraz.htm
