Question

A farmer has 120 feet of fencing available to build a rectangular pen for her pygmy goats. She wants to give them as much room as possible to run.
1.Draw a diagram to represent this problem.
2.Write an expression in terms of a single variable that would represent the area of a rectangle in this family.
3.Find the dimensions of the rectangle with maximum area.
4.What is another name for this kind of rectangle?

Answer 1

Have you number 1?

Answer 2

no :(

Answer 3

Hmmm... you should know what a rectangle looks like...

Answer 4

opposite sides are equal

Answer 5

I do not need to apply any numbers to the sides?

Answer 6

what don't know the numbers....I called the numbers x and y.

Answer 7

but we do know a relationship between x and y

Answer 8

oh alright

Answer 9

we know we have 120 feet of fencing

Answer 10

so you should know the perimeter equals?

Answer 11

yes

Answer 12

so can you state an equation relating x and y (think perimeter)

Answer 13

x+y?

Answer 14

x+y is an expression

I'm looking for an equation that relates x and y (the hint I gave for this equation was to think perimeter )

Answer 15

you have 120 feet amount of fencing
in terms of x and y you have how much amount of fencing ?

Answer 16

how many sides have measurement x?
how many sides have measurement y?

Answer 17

two sides with x and two sides with y

Answer 18

right
x+x+y+y is how much fencing we have
or simplifying
2x+2y is how much fencing we have
but we were also given that we had 120 feet of fencing
so 2x+2y=120

Answer 19

Also you guys probably know the formula for finding the perimeter of a rectangle ...

Answer 20

which is Perimeter=2L+2W

Answer 21

Anyways to make 2x+2y=120 prettier I would divide both sides by 2
giving x+y=60

Answer 22

anyways we have started doing number 2 sorta of
except we need to mention something about the area of this rectangle

Answer 23

what is the area of our rectangle

Answer 24

to find area is length times width

Answer 25

in terms of x and y

Answer 26

our length is ?
and our width is?

Answer 27

so xy?

Answer 28

yes we have Area=xy
I'm just going to use A for area
A=xy

now number 2 also says write in terms of one variable
this is where that perimeter equation we came up with will come in use

Answer 29

remember we had x+y=60
solve for either x or y
(your choice; no need to solve for both though)

Answer 30

x=-y+60

Answer 31

right good
so replace the x in A=xy with x=-y+60
and you will have completed number 2

Answer 32

So A= (-y+60)y?

Answer 33

awesome

Answer 34

Now is this a calculus or algebra class?

Answer 35

if it is calculus we differentiate or you could use algebraic ways because this is a parabola

Answer 36

Pre calculus

Answer 37

have you differentiated in this class?

Answer 38

example: like the derivative of 6x^2 is 12x\

Answer 39

have you done something like that?

Answer 40

no we have not

Answer 41

ok then this is parabola
so you just need to find the vertex of the parabola to find the max

Answer 42

\[f(x)=ax^2+bx+c \\ f(x)=ax^2+\frac{a}{a}bx+c \\ \text{ I multiply second term by } \frac{a}{a} \\ \text{ so it would be easier for you to see how I'm going to factor out } a \\ \text{ from the first two terms } \\ f(x)=a(x^2+\frac{1}{a}bx)+c \\ \text{ so } \frac{1}{a} \cdot b =\frac{b}{a} \\ f(x)=a(x^2+\frac{b}{a}x)+c \\ \text{ now I'm going to leave a space inside the ( ) } \\ \text{ this space will be for completing the square } \\ \text{ I will also mention whatever we add in } \\ \text{ we must subtract out } \\ f(x)=a(x^2+\frac{b}{a}x+?)+c-a? \]
\[\text{ anyways recall } x^2+kx+(\frac{k}{2})^2=(x+\frac{k}{2})^2 \\ \text{ so that means our } k=\frac{b}{a} \\ \text{ so } (x^2+\frac{b}{a}x+(\frac{b}{2a})^2=(x+\frac{b}{2a})^2 \\ \text{ so going back to } f(x) \\ f(x)=a(x^2+\frac{b}{a}x+\frac{b}{2a})^2+c-a(\frac{b}{2a})^2 \\ f(x)=a(x+\frac{b}{2a})^2+c-a(\frac{b}{2a})^2\]

You should recall this is the vertex form of a quadratic.

Where there vertex is:
\[(-\frac{b}{2a},c-a(\frac{b}{2a})^2)\]



you try finding the vertex form (or vertex) of your parabola

Answer 43

At the link are two versions of a solution. They may be helpful.

http://openstudy.com/study#/updates/55035308e4b01982d66287df

@hugy0212