**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

.b. Write an algebraic expression for the Area 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?

f. Create a graph depicting your results.

**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.

a.

b.

c.

d.

e.

f.

**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.

b. Given L as the length of the pen, express the width, W, in terms of L.

c. Write an algebraic expression for the Area of the pen.

d. Where does the maximal area occur? Is there more than one value?

e. Create a graph depicting the results.

**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).

Ex:

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

b. Given L as the length of the pen, express the width, W, in terms of L.

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:

(-2, 6) (5,3) (4,9)

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.

Ex:

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.

Ex:

Which dimensions produced the highest area? Was this value unique?

**Additional Problems:**

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?