Question

A farmer wishes to enclose a pasture that is bordered on one side by a river (so one of the four sides wont require fencing) She has decided to create a rectangular shape for the area and will use barbed wire to create the enclosure there are 600 feet of wire availible for this project and she will use all the wire. what is the maximum area enclosed by the fence then find the maximum of the function.

Answer 1

The area enclosed is (2x)(600-2x).
\[(2x)(600-2x)=1200x-4x^{2}\]
In order to maximize we need the derivative to be equal to 0.
\[y'=1200-8x=0\]
\[1200 = 8x\]
\[x=150\]
Therefore the sides for maximum area are 150*300.

Answer 2

The area is: 45000

Answer 3

whats the maximum of the functions?

Answer 4

Let the sides that are perpendicular to the river be x feet in length
Then the remaining side will be 600-2x in length. (Perimeter = 2x +(600-2x)
Area is length times width or x(600-2x)=600x-2x^2
A=600x-2x^2 dy/dx=-4x+600 set this to = 0 for maxima/minima
-4x+600=0
-4x=-600
x=150 so the dimensions for maximum area or x=150 and 600-(2 * 150) =300
all in feet of course.

Answer 5

You solved for the maximum. x=150

Answer 6

Hi kassia, you are doing pre-calc, right?

So "they" want you to solve this by setting up an equation that represents a parabola, and you say, "OK, the parabola peaks at its vertex, let me find the vertex"
Easier said then done...

Answer 7

yes phi you read my mind

Answer 8

If you're doing pre-calc I apologize. It read like a Calc-1 question I've had before :)

Answer 9

Here's how you start. You have four sides, it's a rectangle, so you opposite sides are equal. So the total perimeter is, for example,
x+y+x+y, where x and y are, say, the length and width.

Answer 10

The equation for the perimeter is
2x + 2y = ??
Help me here, please.

Answer 11

Meanwhile, here's an equation for the area of the rectangle.
Area= xy
Area is length times width.

Answer 12

The amount of fencing is to be used only on three sides and the amount of fencing to be used is 600 feet. this information is to be used to solve this problem.

Answer 13

Oops, thanks. Maybe kassia will fix my mistake??

Answer 14

hey sorry the computer im on is actually retarded but wouldnt it be 2x+y=?? because one of the sides water

Answer 15

hey sorry the computer im on is actually retarded but wouldnt it be 2x+y=?? because one of the sides water

Answer 16

I have to run, but I did work it out step by step for kassia.

Answer 17

hey sorry the computer im on is actually retarded but wouldnt it be 2x+y=?? because one of the sides water

Answer 18

kjkl

Answer 19

kjkl

Answer 20

kjkl

Answer 21

kjkl

Answer 22

Yes, 2x + y= ?, but the next step is what goes in the place of the question mark? we want an equation. hint, we are very close.

Answer 23

The delay is annoying. So let's forge ahead....
the equation for the perimeter is
2x+y= 600

Area= x y

Next step is to replace y in the Area equation, so there are only x's. Why, well because you have done so many of these problems you just know!

Answer 24

Hi Kas, Did you give up? Or did you just want the answer w/o knowing all the details?

Answer 25

oh sorry i got distracted alright so should i plug in 3 for y?

Answer 26

Solve for y using the perimeter equation:
2x+y= 600

y= ??

Answer 27

but i cant solve for y if i dont have x

Answer 28

By solve, I mean just use algebra so y is by itself on one side of the equal sign, and everything else is on the other side. True we don't know what x is,but that's ok.

Answer 29

so y=300-x or y=600-2x

Answer 30

To solve for y, subract 2x from both sides
2x+y-2x= 600 - 2x
On the left, the 2x-2x cancel, so you get y= 600 - 2x
So your 2nd answer is good, the first not so good. As a boring aside, notice
if we start with y= 600-2x and divide both sides by 2 you get
y/2 = 300 -x (but that's not what we want.

Next step, in the equation
Area= x y
replace y with its equivalent 600-2x. Then multiply it out.

Answer 31

\[2x^2+600x\]

Answer 32

So you should get a new equation
Area= x(600-2x)
Area = 600x - 2 x^2
or, rearranging
\[A= -2x ^{2}+600x\]

Did you get this?

Answer 33

yeah so then should i take out a 2x so i get 2x(-x+300) and then can i just set both equations equal to zero?

Answer 34

factor out 2x would be going backwards. What you should be doing is looking at the equation for area
A= -2 x^2 + 600x
and it should remind you of the equation for a parabola (because surprise!) it is.

Answer 35

okay so whats the next step then?

Answer 36

First, I just noticed you left off the minus sign off x^2 I hope that was just a typo. other wise you'll have to practice multiplying. The trouble with math is a tiny mistake ruins everything.

OK, you have an equation that tells you the area if you know one side of the fence (labelled x). So the area you get if the side has zero length is 0. if x=1,
Area is -2*1^2 +600(1)
= -2 + 600 = 598

You get the idea? But we know this is a parabola, and it has a peak at its vertex
A , the area, peaks at the parabola's vertex.

Answer 37

So the next step is to find the x value of the vertex. Do you remember from yesterday?

the vertex is -b/(2a)
for the parabola
y=a x^2 +b x +c

Answer 38

Did I confuse you??

Answer 39

Once you find the x value of the vertex, use that x value in the equation for the Area
Area= -2 x^2 + 600 x
That's going to be the maximum area you can enclose with your fence.

Answer 40

If everything works out, we should get the same answer as from Calculus.

Answer 41

got it thanks!

Answer 42

Excellent! Can you post your work?