hugy0212
  • hugy0212
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?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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freckles
  • freckles
Have you number 1?
hugy0212
  • hugy0212
no :(
freckles
  • freckles
Hmmm... you should know what a rectangle looks like...

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More answers

freckles
  • freckles
|dw:1443890804767:dw| opposite sides are equal |dw:1443890824575:dw|
hugy0212
  • hugy0212
I do not need to apply any numbers to the sides?
freckles
  • freckles
what don't know the numbers....I called the numbers x and y.
freckles
  • freckles
but we do know a relationship between x and y
hugy0212
  • hugy0212
oh alright
freckles
  • freckles
we know we have 120 feet of fencing
freckles
  • freckles
so you should know the perimeter equals?
hugy0212
  • hugy0212
yes
freckles
  • freckles
so can you state an equation relating x and y (think perimeter)
hugy0212
  • hugy0212
x+y?
freckles
  • freckles
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 )
freckles
  • freckles
you have 120 feet amount of fencing in terms of x and y you have how much amount of fencing ?
freckles
  • freckles
how many sides have measurement x? how many sides have measurement y?
hugy0212
  • hugy0212
two sides with x and two sides with y
freckles
  • freckles
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
freckles
  • freckles
Also you guys probably know the formula for finding the perimeter of a rectangle ...
freckles
  • freckles
which is Perimeter=2L+2W
freckles
  • freckles
Anyways to make 2x+2y=120 prettier I would divide both sides by 2 giving x+y=60
freckles
  • freckles
anyways we have started doing number 2 sorta of except we need to mention something about the area of this rectangle
freckles
  • freckles
what is the area of our rectangle
hugy0212
  • hugy0212
to find area is length times width
freckles
  • freckles
in terms of x and y
freckles
  • freckles
our length is ? and our width is?
hugy0212
  • hugy0212
so xy?
freckles
  • freckles
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
freckles
  • freckles
remember we had x+y=60 solve for either x or y (your choice; no need to solve for both though)
hugy0212
  • hugy0212
x=-y+60
freckles
  • freckles
right good so replace the x in A=xy with x=-y+60 and you will have completed number 2
hugy0212
  • hugy0212
So A= (-y+60)y?
freckles
  • freckles
awesome
freckles
  • freckles
Now is this a calculus or algebra class?
freckles
  • freckles
if it is calculus we differentiate or you could use algebraic ways because this is a parabola
hugy0212
  • hugy0212
Pre calculus
freckles
  • freckles
have you differentiated in this class?
freckles
  • freckles
example: like the derivative of 6x^2 is 12x\
freckles
  • freckles
have you done something like that?
hugy0212
  • hugy0212
no we have not
freckles
  • freckles
ok then this is parabola so you just need to find the vertex of the parabola to find the max
freckles
  • freckles
\[f(x)=ax^2+bx+c \\ f(x)=ax^2+\frac{a}{a}bx+c \\ \text{ I multiply second term by } \frac{a}{a} \\ \text{ so it would be easier for you to see how I'm going to factor out } a \\ \text{ from the first two terms } \\ f(x)=a(x^2+\frac{1}{a}bx)+c \\ \text{ so } \frac{1}{a} \cdot b =\frac{b}{a} \\ f(x)=a(x^2+\frac{b}{a}x)+c \\ \text{ now I'm going to leave a space inside the ( ) } \\ \text{ this space will be for completing the square } \\ \text{ I will also mention whatever we add in } \\ \text{ we must subtract out } \\ f(x)=a(x^2+\frac{b}{a}x+?)+c-a? \] \[\text{ anyways recall } x^2+kx+(\frac{k}{2})^2=(x+\frac{k}{2})^2 \\ \text{ so that means our } k=\frac{b}{a} \\ \text{ so } (x^2+\frac{b}{a}x+(\frac{b}{2a})^2=(x+\frac{b}{2a})^2 \\ \text{ so going back to } f(x) \\ f(x)=a(x^2+\frac{b}{a}x+\frac{b}{2a})^2+c-a(\frac{b}{2a})^2 \\ f(x)=a(x+\frac{b}{2a})^2+c-a(\frac{b}{2a})^2\] 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
Directrix
  • Directrix
At the link are two versions of a solution. They may be helpful. http://openstudy.com/study#/updates/55035308e4b01982d66287df @hugy0212

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