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!!

The Giver Government

I already know that the society in the Giver is a bad one, but what about the government system?  Does it meet the standards that every government should?  Is is protecting the people and the place itself?

Every government has a set of expectations that should be met.  Governments should protect the people from outside threats, and those inside.  Part of doing this is deciding who enters and leaves the territory or country.  If any government was to let anyone in under any circumstances and didn't guard their borders, they would be overthrown very quickly.  Armies could get inside without the government knowing it.  And as for leaving, if a criminal escaped from prison, he or she would head strait for the border and cross it.  In doing this, the criminal would have escaped justice and could continue to do things like his past crimes without punishment.  This connects to the government needing a force of men to fight in wars or keep laws.  If there wasn't any army, the country would be crushed by nearby societies, and there would be no stopping them.  If there were no police, people could get away with anything they want.  The government also needs to meet the people's needs, such as food, shelter, water.  If the people of a country don't have their basic needs, there won't be many people who can contribute to society because they would be so weak.  However, one of the most important reasons for a government, is to to prevent conflict and establish peace.  If there was no government, everyone would be separated by families.  Blood feuds would happen easily, and families could kill each other and no one would do anything about it.

Thinking back to the community of the Giver, were all these requirements met?  If you think about it, yes, they were.  The people in the community were protected completely.  For one, there was no pain, because the government made anti-pain shots and meds.  There were no outside forces in the book, and in the movie they were above the clouds on a giant plateau.  No one committed any crimes because they were brainwashed, and they could think of doing something against the law.  This means no murder and no thievery.  Does the government have the means to fight off a war?  They don't have armies, but with how advanced in technology they are, they probably have missiles, or an army of robots!  Also, in the beginning of the book, they say they have the means to shoot down a jet, so they could easily launch the rocket ,or whatever they would shoot the jet with, at an army of soldiers marching to destroy the community.