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Mathematical Literacy for Dairy Farm Management

Background: During our field to trip Gilbert and Sons Dairy Farm, we saw several examples of mathematics that are essential to managing a profitable dairy farm. In particular, the farmer must be able to plan and construct physical equipment like fencing and barns to house animals, calculate proportions to prepare the daily food rations for the cattle, and optimization techniques to maximize profits.

Goals: Excel/GSP proficiency, Maximization of Areas, and expressing aspects of the physical world algebraically.

For this series of activities, we will be considering the mathematical planning involved in constructing fencing pens to hold cattle. The first activities will concern the materials involved in constructing fences out of metal wiring. The fence needs to be constructed subject to the following parameters:

o The fence will be divided into separate partitions, consisting of solid metal posts and connected by metal wiring.
o The solid posts are 4 inches in diameter.
o The fence portion between the posts will be constructed out of metal wiring.
o There will be parallel wiring, 4 inches apart, with the lowest wiring located 4 inches off the ground, and the highest wiring located at the top of the fence.
o The hypotenuse of the fence must be reinforced with a wire, and will extend from the top corner of a particular portion of fencing to the bottom rung of wire at the adjacent post.
o The wiring will be continuous around the perimeter of the fence for the respective parallel heights. The wiring of the diagonal portion will be added after the parallel wiring is laid.
o The wiring is wrapped around the post once time at each respective height. The diagonal wiring is wrapped around the post once at each end as well.1. To ensure understanding of the structure of a fence, sketch A) a diagram of one section of a fence and B) a diagram of two adjacent fencing partitions. Assume that the dimensions of the partitions are 3 feet tall and 4 feet wide.

2. a. Consider a single partition of the fencing that is x feet high. How many parallel ‘tiers’ of wiring will be required? Write this result as an algebraic expression.

b. Consider a single partition of the fencing that is y feet long. How long will the respective parallel ‘tiers’ of wiring be? Remember to include the wiring around the fence posts on the ends. Write this result as an algebraic expression.

c. Can you develop a formula for the number of posts required to build ‘n’ linearly linked fence partitions? Write this formula as an algebraic expression.d. Develop a formula for the number of posts needed to construct a square pen, with ‘n’ partitions of fencing on each side. Write this formula as an algebraic expression.

e. If you construct three connected fence partitions, with the dimensions between the posts being 4 feet wide and 3 feet high, how much wiring will you require, and how many posts will be used? Write separate algebraic expressions for the wiring required for a particular partition ‘between the posts’, and a separate expression depicting the wiring required to wrap around the posts.

3. a. Given that metal wiring costs $2 per foot, and metal posts costs $5 each, how much would it costs to create a square fenced area, with 3 foot (high) by 4 foot (wide) wiring partitions with 7 partitions of fence on each side?

b. What is the largest square fenced region (3x4) with posts that you can construct with $ 500,000?

Now that we have a sound understanding of the materials required to construct our animal pens, we must now decide what shapes and dimensions produce the maximal area for a fixed amount of fencing. To this end, the following series of questions will involve investigating fencing dimensions with a fixed perimeter and the resulting areas of various geometric configurations. The following calculations will require Microsoft Excel and Geometer’s Sketchpad.

1. Russ has 50 feet of steel fencing, made from 50 1-foot sections. He is going to construct a rectangular stable for calves. Some natural questions to ask are: What are all possible dimensions of this garden? Would the possible dimensions of this stable produce the same area?
Create a spreadsheet with Microsoft Excel to investigate with data.
Ex:

2.a. If the length of the fencing area is represented by L, write an algebraic expression for the Width of the fencing region

c. Do some values for length and width produce the same area? If so, which ones?

d. Is it possible to produce a fenced area with square dimensions? If so, what are the dimensions?

e. What was the maximum value that you found in your table? Did this value occur more than once?

3. Answer the questions from question 2, now given 100 feet of fencing. Again, we will be investigating rectangular regions using Excel to elucidate the answers.

4. Let us suppose that we wish to again make a rectangular region with 100 feet of fencing, but we need the fenced area partitioned into two areas (not necessarily equal in area).
What dimensions of fenced region are possible to construct? List a few possible dimensions.
Ex:

a. Create a 3-column excel spreadsheet to represent the possible dimensions.

5. We now wish to consider creating a rectangular pen with 100 feet of fencing, but this time we want three separate areas within the pen (not necessarily of equal area).

c. Write an algebraic expression for the area of the pen.d. Where does the maximum area occur? Is there more than one value?

6. Let us now consider triangular regions of fencing using GSP and Excel.
We wish to know what possible triangular areas we can enclose with 100 feet of fencing. However, it is difficult to calculate the area of the resulting triangles, as our only ‘known’ formula for the area of a triangle is (1/2)BH, which assumes knowledge of the altitude of our respective triangles. We will now investigate ways to get around this problem.

Open GSP, and go to the graph menu, and select ‘show grid’. Plot the following points using ‘plot points’ under the graph menu:

Select the three points, and choose ‘construct segments’ under the construct menu. Next, select the three points and choose ‘construct triangle interior’. With the interior selected, go to the measure menu, and measure the area of the triangle.

We now wish to examine the number of lattice points, (ie, points on the integer coordinate grid) on the interior of the triangle (interior points), as well as on the border (boundary points). Record these values as well as the area.

Now choose 5 triplets of points, and repeat the above process for each triplet. We are going to create an Excel file.

Now choose 3 quartets of points, and create the resulting quadrilaterals, recording the number of Interior Points, Boundary Points, and Area as in the above chart.

Repeat the process, choosing 2 quintets of points, and adding these results to the chart above.

Can you derive any general formula for the Area of a lattice polygon given the number of lattice points on its interior and on the boundary?

7. Returning to the notion of triangular fencing regions, we would like to create a chart of the possible dimensions and the resulting area, but we need an explicit algebraic formula to achieve this goal.

We are going to use Heron’s Formula, which states that A, the area of a triangle with sides a, b, and c, is equal to the following:

Where s = (a + b + c)/2

Create an excel chart depicting the possible values you can create with 100 feet of fencing, and use our new formula to compute the area of the resulting triangles. For clarity, let us consider only isosceles triangles for now.

Prove:

Suppose a goat is tethered in a square fenced pasture from one of the corners of the fence. How long should the tether be so that the goat can graze over exactly half of the field?

Consider a problem similar to the one above, but where the goat is tethered to the midpoint of one of the sides of the square. How long should the tether be to enable the goat to graze over exactly half of the field?