Calculus 2 Help-A rancher has property along a river that he needs to divide into three separate,
equal-sized regions: one for the cows, one for the horses, and one for the sheep. The
rancher has 1000 ft of available fencing with which to divide up the land. There will need
to be enough fencing for three outer sides of the property as well as two lengths of
fencing to divide the region into three parts. Determine the values of x and y, and the
area of an individual pen so that its area is maximized. Be sure to use calculus to
justify you have found a maximum.

- anonymous

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- anonymous

So i got that the equation will be F=1000ft=4x+y and A=1/3xy

- amistre64

Is the river included as one side of the pens? cause in alot of problems it is..

- anonymous

you dont need fence for the river part

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- anonymous

so no

- amistre64

then we need 4x + y = 1000, right?

- anonymous

yea the 2 equations i set up are my up top

- amistre64

good, but I need to work it thru in my head too:)

- anonymous

no prob

- anonymous

take your time

- amistre64

and the area of each pen is x(y/3)

- amistre64

Since y = 1000 - 4x, can we use that in the area?

- anonymous

yea perfect

- amistre64

the deriviate and solve for max right?

- amistre64

esxucse typos, my keyboard hates me:)

- anonymous

no prob, yea

- amistre64

i get (3000 - 24x)/9 for a derivative...what about you?

- anonymous

hold on, did you set it up first by substituting?

- amistre64

x = 3000/24 = 1500/12 = 750/6.....
yeah substitute in y=1000-4x.

- anonymous

so A(x)=1000x-4X^2 all over 3

- amistre64

x = 375/3...x = 125 for me

- anonymous

sorry im going slow

- amistre64

its ok...gives me time for my mind to wander :)

- anonymous

was my A(x) correct?

- amistre64

Your A(x) was correct...

- anonymous

then you took the derivative?

- amistre64

yep, the derivative tells us where we can find bends inthe graph that are our max and min values.

- amistre64

You remember your quotient rule?

- anonymous

yea let me do it one sec

- anonymous

3000-24x all over 9

- amistre64

yep; and theres another way we could have approached it. 1/3 is a constant so we can pull it out and just derivative (1000x-4x^2). Either way is the same

- anonymous

y did i get x=27000/24

- amistre64

solve for y' = 0.
What makes (3000-24x)/9 = 0?

- amistre64

the top has to equal 0 for ay of it to matter; so throw away the 9

- amistre64

3000-24x = 0 what is x?

- anonymous

so 1000-8x

- amistre64

thats good too

- amistre64

1000 - 8x = 0...is the same thing

- anonymous

1000/8=x

- anonymous

125=x

- amistre64

yep, now reduce it or use it as is if you want :)
Good job

- anonymous

y=1000-4(125)

- amistre64

yep....

- anonymous

A=1/3(500)*125

- amistre64

(500)(125)/3 is more accurate, but yeah...

- anonymous

20833ft^2, that seems large?

- amistre64

when talking about football field measurements, I imagine it would be :)

- anonymous

oh ok , now the other part is the maximum

- anonymous

2nd derivative right?

- amistre64

total inclosed area is 62500 ft^2

- amistre64

20833 1/3 each

- anonymous

they just want individual

- anonymous

so 20833ft^2 per pen

- amistre64

we already found the maximum... there is only one hump in a quadratic..

- amistre64

you can second derive it if you want, just to prove it to yourself, but it aint needed

- anonymous

i need to do all steps she takes off for everything

- anonymous

Is this 100% right, she had like inequality equations and domain stuff on her examples. I am confused

- amistre64

lol..when y'' is (+) we have a concave down which is an upside down bowl on the graph. And that is the max.

- amistre64

its right....so far :) use the math luke.....the maaatthhhh...

- anonymous

k haha

- amistre64

might wanna double check that concavity thing, I never use it enough to remember if Im right :)

- anonymous

soundds good

- anonymous

it should be negative to be a max
<0

- amistre64

remember CAVE....CAVE UP is (+) and CAVE DOWN is (-) or the other way around..

- anonymous

i mean in concavity relation

- anonymous

if its >0, its concave up oh okay

- amistre64

lol....ill trust you on that :)

- anonymous

how do you have a max then

- amistre64

picture the graph of (-x^2) thats all this is.

- anonymous

oh okay

- anonymous

hold up is the second derivative -8

- amistre64

thats right, so the cave is down. its an upside down bowl.

- amistre64

-(x^2) ... y'=-(2x)....y'' = -(2)

- anonymous

oh i see

- anonymous

sorry, lol i am so bad at optimization

- amistre64

:)

- anonymous

how do the domains work

- anonymous

she wants us to show that too D:(0, )

- amistre64

there is nothing limiting our domain, so it canbe any real number (R)

- amistre64

look at the fractions we made, if anything makes the bottom a 0, throw it out. Nothing matters, so keep it all

- anonymous

so (0, Infinity)

- amistre64

I suppose we should focus on positive numbers to make sense of it all, so yeah. except it should be [0,inf) 0 is included.

- amistre64

wait....just had athought

- amistre64

we only have so much fencing to work with... so make the D [0,1000]

- amistre64

or you might wanna think about like this: we need 4 amounts of x so ...maybe...[0,1000/4] ? take a gander at it and see :)

- anonymous

she likes simplicity

- amistre64

my gut tells me that we do [0,1000]..

- anonymous

my friend said the domain is 0 to 250

- amistre64

does your friend tell you how to find the max area :)

- anonymous

its your x

- anonymous

He said solve 1000x-4x^2 for x

- anonymous

and thats your A( )>0

- anonymous

which is your max by the Extreme value theorem

- amistre64

thats fine.....well use 0 to 250......its as good as anyother I can think of :)

- anonymous

no he said solve the inequality 1000-4x>0

- anonymous

thats what the 250 is

- amistre64

1000/4 = 250....i know iknow...yeahyeahyeah....

- anonymous

can we walk through it step by step, I am so confused

- anonymous

is 125 okay because it falls in the domain?

- amistre64

125 is good

- amistre64

step 1: we are given certian rules that govern our equations...did we use them right?

- anonymous

yes

- anonymous

A'' in that case is <0 and negative which is why its a max?

- amistre64

we Know: 4x + y = 1000 and we KNow : Area per pen = xy/3

- anonymous

yea perfect

- amistre64

yes, since it is <0 it is a max

- amistre64

we want max area per pen so we substitute y in our area formula and get the derivative.

- amistre64

when y' = 0 we have a bend, a max OR a min.

- anonymous

which we just say is 1000-8x

- amistre64

yes.... and y'' tells us how the graph behaves at certain points.
(-) = CAVE DOWN = max point
(+) = CAVE UP = min point
(0) = point of inflection.

- anonymous

so y''= -8

- amistre64

tells us it is always concave DOWN becasue its always (-8)

- amistre64

you still confused? or feeling more confident?

- anonymous

yea im getting it a little. He said since x fits in the open interval, you take the 2nd derivative

- anonymous

ahhhh where do I plug in our x in our second derivative?

- amistre64

why would you need to pluu it in? the only reason to plug in an x in the secondD is to figure out where is changes sign. -x^2 never changes concavity so it is always, and I mean ALWAYS cavedown

- anonymous

yea

- amistre64

x^3 changes....so that youd check :) but we aint messin with no x^3

- anonymous

what is A'' of (1000x-4x^2)/3

- anonymous

lets see if it works anyway

- amistre64

A' = (1000-8x)/3
A'' = -8/3
A''' = 0

- anonymous

thats what i thought

- amistre64

you say this is calc 2 stuff?

- anonymous

yea why?

- amistre64

cause im only in trig at the moment.....and it is sooooo boring there.

- anonymous

haha wow

- anonymous

how are you helping me so well then?

- amistre64

i dunno, maybe I is smart ? I taught myself calculus in intermediate algebra to pass the time....

- anonymous

haha nice yea this is college calc 2

- amistre64

I still need alot of practice with integrals tho..... and word problems. I loathe word problems :)

- anonymous

yea you are still decent

- amistre64

Im gonna get my masters in math...and teach college math

- amistre64

:) thanx

- anonymous

i bet you can do it! easy 4 u

- amistre64

easy as....should I?......PI lol

- anonymous

lol

- anonymous

thx so much though

- amistre64

no prob... keep up the good work :)

- anonymous

my friend confirmed everything

- anonymous

its right

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