Agriculture

The shift from small, independent family farms to fewer, larger farms that rely heavily upon capital and technology began in 1793 with the invention of the cotton gin. The cotton gin enabled a single farm worker to clean 1,000 pounds of cotton in a day—a tremendous increase from the 50 pounds daily that an individual could clean without the gin. The continual adoption of new technologies in agriculture, along with the application of genetics, chemistry, physics, biology, and plant and animal nutrition, has created an agricultural industry in which mathematics play an increasingly important role.

Crop Production Planning Models

Because most soils and climates are suitable for producing a variety of agricultural products, many farmers build a mathematical model to determine what crops to plant in which fields. Whereas some costs included in the model, such as insurance, taxes, depreciation, interest, land rents, and loan payments, are fixed, others vary widely, depending on the crop selected, the amount of labor and chemical to be applied, and other factors that can often be controlled by the farmer.

For example, corn seed is commonly sown at rates ranging from 19,000 to 32,000 seeds per acre. Because closely spaced plants compete for sunlight, water, and nutrients, excessive populations result in lower yield per plant. However, this decrease may be offset by an increase in total crop yield per acre. The line graph shows the effect of plant population on yield per acre of one particular corn hybrid.

Fertilizer selection requires calculations of how much of each nutrient is needed for the various crops. Nitrogen, for example, is available in solid, liquid, and gaseous man-made chemical fertilizers, as well as in organic fertilizers such as manure. Because each of these compounds contains a different percentage of nitrogen and because many fertilizers supply more than a single nutrient, the farmer's analysis of fertilizer costs and benefits often includes complicated algebraic equations.

After calculating projected expenses for the production of each crop, the farmer builds the revenue side of the model. After determining the area of

each field, the farmer then figures anticipated yield per acre, using weather expectations and historical data from the various fields. Acres, yield, and expected selling price are multiplied together to estimate revenue for each potential crop.

Careful consideration of these projected income and expense figures allows the farmer to determine which crops have the greatest likelihood of returning a profit and to plant accordingly. This exercise must be repeated each growing season for every field, as input costs and anticipated sales prices fluctuate.

Mathematics in the Field

Once crop selections have been made, farmers begin the processes of planting, nurturing, and harvesting farm commodities. These activities are likely to require the use of mathematics daily.

For example, to limit their effect on the environment, farmers use low application rates for fertilizers, herbicides, and insecticides. Typical application rates range from 0.5 to 16 ounces of active ingredient per acre. The chemicals are diluted with water for application. Mathematical formulas, such as the one below, are used to obtain the proper dosage through simultaneous adjustments of spray pressure, tractor speed, and chemical concentration.

This is just one of many commonplace examples of the use of math in producing a crop. Farmers who irrigate must calibrate irrigation-pump engine speed to apply a given amount of water to a given area of cropland. Corn growers figure time remaining until harvest by calculating growing degree days, which is a measure of heat units needed by a corn plant to reach maturity. Formulas relating relative humidity and time to moisture content are used by farmers who must dry alfalfa or other forage crops before baling

them as hay. The opportunities to use math in crop production are nearly endless.

Livestock Production

Decreasing profit margins and increasing demand for uniform, safe animal food products have boosted the significance of mathematics in livestock operations. Cattle, poultry, hog, and sheep producers keep individual productivity records on each animal, retaining for breeding purposes only those that efficiently convert feed into pounds of meat, milk, eggs, or wool.

For each animal, producers track rates of gain, feed consumption, carcass fat percentage, ratios of both more expensive and less desirable cuts of meat, and other data. Cost-effective culling and breeding decisions are made based on these numbers and formulas.

Additionally, livestock producers use algebraic formulas to create animal feed rations that are nutritionally balanced and cost effective. Feed components such as corn, alfalfa hay, oats, cottonseed hulls, and molasses all supply varying amounts of the micronutrients necessary for efficient livestock production. Complete rations are mixed using a combination of these and other ingredients. Careful attention to the balancing of rations not only controls costs but also increases productivity per animal.

Marketing

Many agricultural commodities can either be sold at harvest or stored for future sale. To estimate the amount of grain in storage and available for sale, geometric equations such as those in the table are used.

The decision of whether to sell or retain the harvest is essentially a mathematical exercise. Factors to consider include the cost of constructing on-farm

EQUATIONS FOR DETERMINING VOLUME OF STORED GRAIN
All measurements are in feet
Rectangular Bin	Bushels=0.8036 (length)(width)(average depth of grain)
Round Bin	Bushels=0.8036 (3.1416)(radius of base squared)(average depth of grain)
Piled grain (conical shaped pile)	Bushels=0.8036 (1.0472)(radius of base squared)(height of pile)

storage facilities or renting storage space from a local grain elevator, the potential increase in sales revenue if the product is sold at a higher price, the interest that could be earned if the crop were sold and money banked at harvest, and the risk involved if the price should decline or if part of the crop is lost as a result of spoilage.

Forward contracting, or locking in a sale price for future delivery, is done through futures exchanges such as the Chicago Board of Trade. Marketing through a futures exchange requires the farmer to use charts and graphs depicting market trends, trading volumes, moving averages, and other relevant data.

Mathematics and Agricultural Productivity

In countries where little attention is given to the mathematical and scientific aspects of farming, up to 70 percent of the population earns its livelihood through agriculture. As math and science are diligently applied to agriculture, productivity increases so that fewer farmers are needed. Thus, the number of farmers in the United States has declined from nearly 7 million in the 1930s to less than 2 million, or under 3 percent of the population.