Goat Pen Math

In math class we were handed certain problems to solve and then explain.  Here's mine:

You have 40 meters of rectangular fencing material and you want to build a rectangular pen for your goats, but you are not sure what size to make the dimensions.  (You only know that you will use all 40 meters of fence.)

a.  What will the perimeter of the fence be?
This is an easy question.  It basically said the answer in the up at the top.
You have 40 meters of rectangular fencing...
Perimeter the length of all the sides combined, laid out in a straight line.  This first part means that the same thing, all the fencing in a line is 40 meters long.  Therefore, the answer to a. is 40.

b. Draw a diagram and label the dimensions using x for the width.
(there's also a hit, but we're too cool for hints ;))

So, this means that we've got to draw a rectangle with the width labeled x.  


Of course, it doesn't have to be red, I just wanted it to be the same color as the one explaining stuff.
And while were sitting here wondering what the length is, let's find out how to find it, no matter the dimensions.
Let's use y as the length

20 - x = y

Since all the lengths are 40, that makes two sides equal 20. 20 minus the width must equal the length, if 20 is both of them added together.

c. Make a table of values (using only integers) to see how the width of you pen affects the area, and graph these values

A table of values is a table where each value matches with another.  In our case, these values are the length and width of your goat pen.

x	y
1	19
2	18
3	17
4	16
5	15
6	14
7	13
8	12
9	11
10	10
11	9
12	8
13	7
14	6
15	5
16	4
17	3
18	2
19	1

And when we graph these points, they look a little like this:

d. From your graph, determine what dimensions would give you the largest area.

This is simple multiplication.  Right in the middle, there's 10x10, which is 100.  Next to it, is 11x9, which is 99.  The product gets lower the farther you go from (10,10).  Commutative property, so same deal on the other side.

So basically , you've gotta make your goat pen a whopping 10 by 10 meters.

e. Write an equation for the area  of your pen, using your diagram.

The most commonly used equation for area is lw = A

But that's the lame generic equation.  We want the new+improved awesome equation with the right numbers.

So, it could be A = x(20-x), because that generates our length in the parentheses, and has the x multiplying it.  However, it can still be simplified.  If we distribute the x, it turns into A = 20x - x².  

f. Graph your equation using a graphing calculator and compare it to the graph you drew*.

we''re rebels so we did it digitally and didn't draw it, but we're still cool.

So this is what you get when you type in your equation into a graphing calculator.

 What do you make of this meaningless line?  Well, if you zoom out...

 You eventually get two lines!  They look like they will intersect.  But if you zoom out,

It turns out that it's a huge curve!  At the climax is the number 100.  What's the area of your goat pen?  That's right, 100.

Now let's compare this curve to our other graph (the one we "drew").

I don't know what's similar about these, execpt for something you can't see in this image.  The 10x10 mark on the bottom lines up with the climax of the big curve.  I think this means our previous graph was correct.

Now we have to make a graph out that big curve.

Now you can see that the area starts low, and gets higher and higher, until it reaches the maximum (100), and then goes back down, less and less.  Just like our areas we calculated higher up.

I hope this helped you understand how to come up with equations to make graphs to find areas.

Now go make your goat pen!!