anonymous
  • anonymous
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.
Mathematics
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schrodinger
  • schrodinger
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anonymous
  • anonymous
So i got that the equation will be F=1000ft=4x+y and A=1/3xy
amistre64
  • amistre64
Is the river included as one side of the pens? cause in alot of problems it is..
anonymous
  • anonymous
you dont need fence for the river part

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anonymous
  • anonymous
so no
amistre64
  • amistre64
then we need 4x + y = 1000, right?
anonymous
  • anonymous
yea the 2 equations i set up are my up top
amistre64
  • amistre64
good, but I need to work it thru in my head too:)
anonymous
  • anonymous
no prob
anonymous
  • anonymous
take your time
amistre64
  • amistre64
and the area of each pen is x(y/3)
amistre64
  • amistre64
Since y = 1000 - 4x, can we use that in the area?
anonymous
  • anonymous
yea perfect
amistre64
  • amistre64
the deriviate and solve for max right?
amistre64
  • amistre64
esxucse typos, my keyboard hates me:)
anonymous
  • anonymous
no prob, yea
amistre64
  • amistre64
i get (3000 - 24x)/9 for a derivative...what about you?
anonymous
  • anonymous
hold on, did you set it up first by substituting?
amistre64
  • amistre64
x = 3000/24 = 1500/12 = 750/6..... yeah substitute in y=1000-4x.
anonymous
  • anonymous
so A(x)=1000x-4X^2 all over 3
amistre64
  • amistre64
x = 375/3...x = 125 for me
anonymous
  • anonymous
sorry im going slow
amistre64
  • amistre64
its ok...gives me time for my mind to wander :)
anonymous
  • anonymous
was my A(x) correct?
amistre64
  • amistre64
Your A(x) was correct...
anonymous
  • anonymous
then you took the derivative?
amistre64
  • amistre64
yep, the derivative tells us where we can find bends inthe graph that are our max and min values.
amistre64
  • amistre64
You remember your quotient rule?
anonymous
  • anonymous
yea let me do it one sec
anonymous
  • anonymous
3000-24x all over 9
amistre64
  • 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
  • anonymous
y did i get x=27000/24
amistre64
  • amistre64
solve for y' = 0. What makes (3000-24x)/9 = 0?
amistre64
  • amistre64
the top has to equal 0 for ay of it to matter; so throw away the 9
amistre64
  • amistre64
3000-24x = 0 what is x?
anonymous
  • anonymous
so 1000-8x
amistre64
  • amistre64
thats good too
amistre64
  • amistre64
1000 - 8x = 0...is the same thing
anonymous
  • anonymous
1000/8=x
anonymous
  • anonymous
125=x
amistre64
  • amistre64
yep, now reduce it or use it as is if you want :) Good job
anonymous
  • anonymous
y=1000-4(125)
amistre64
  • amistre64
yep....
anonymous
  • anonymous
A=1/3(500)*125
amistre64
  • amistre64
(500)(125)/3 is more accurate, but yeah...
anonymous
  • anonymous
20833ft^2, that seems large?
amistre64
  • amistre64
when talking about football field measurements, I imagine it would be :)
anonymous
  • anonymous
oh ok , now the other part is the maximum
anonymous
  • anonymous
2nd derivative right?
amistre64
  • amistre64
total inclosed area is 62500 ft^2
amistre64
  • amistre64
20833 1/3 each
anonymous
  • anonymous
they just want individual
anonymous
  • anonymous
so 20833ft^2 per pen
amistre64
  • amistre64
we already found the maximum... there is only one hump in a quadratic..
amistre64
  • amistre64
you can second derive it if you want, just to prove it to yourself, but it aint needed
anonymous
  • anonymous
i need to do all steps she takes off for everything
anonymous
  • anonymous
Is this 100% right, she had like inequality equations and domain stuff on her examples. I am confused
amistre64
  • 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
  • amistre64
its right....so far :) use the math luke.....the maaatthhhh...
anonymous
  • anonymous
k haha
amistre64
  • amistre64
might wanna double check that concavity thing, I never use it enough to remember if Im right :)
anonymous
  • anonymous
soundds good
anonymous
  • anonymous
it should be negative to be a max <0
amistre64
  • amistre64
remember CAVE....CAVE UP is (+) and CAVE DOWN is (-) or the other way around..
anonymous
  • anonymous
i mean in concavity relation
anonymous
  • anonymous
if its >0, its concave up oh okay
amistre64
  • amistre64
lol....ill trust you on that :)
anonymous
  • anonymous
how do you have a max then
amistre64
  • amistre64
picture the graph of (-x^2) thats all this is.
anonymous
  • anonymous
oh okay
anonymous
  • anonymous
hold up is the second derivative -8
amistre64
  • amistre64
thats right, so the cave is down. its an upside down bowl.
amistre64
  • amistre64
-(x^2) ... y'=-(2x)....y'' = -(2)
anonymous
  • anonymous
oh i see
anonymous
  • anonymous
sorry, lol i am so bad at optimization
amistre64
  • amistre64
:)
anonymous
  • anonymous
how do the domains work
anonymous
  • anonymous
she wants us to show that too D:(0, )
amistre64
  • amistre64
there is nothing limiting our domain, so it canbe any real number (R)
amistre64
  • amistre64
look at the fractions we made, if anything makes the bottom a 0, throw it out. Nothing matters, so keep it all
anonymous
  • anonymous
so (0, Infinity)
amistre64
  • 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
  • amistre64
wait....just had athought
amistre64
  • amistre64
we only have so much fencing to work with... so make the D [0,1000]
amistre64
  • 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
  • anonymous
she likes simplicity
amistre64
  • amistre64
my gut tells me that we do [0,1000]..
anonymous
  • anonymous
my friend said the domain is 0 to 250
amistre64
  • amistre64
does your friend tell you how to find the max area :)
anonymous
  • anonymous
its your x
anonymous
  • anonymous
He said solve 1000x-4x^2 for x
anonymous
  • anonymous
and thats your A( )>0
anonymous
  • anonymous
which is your max by the Extreme value theorem
amistre64
  • amistre64
thats fine.....well use 0 to 250......its as good as anyother I can think of :)
anonymous
  • anonymous
no he said solve the inequality 1000-4x>0
anonymous
  • anonymous
thats what the 250 is
amistre64
  • amistre64
1000/4 = 250....i know iknow...yeahyeahyeah....
anonymous
  • anonymous
can we walk through it step by step, I am so confused
anonymous
  • anonymous
is 125 okay because it falls in the domain?
amistre64
  • amistre64
125 is good
amistre64
  • amistre64
step 1: we are given certian rules that govern our equations...did we use them right?
anonymous
  • anonymous
yes
anonymous
  • anonymous
A'' in that case is <0 and negative which is why its a max?
amistre64
  • amistre64
we Know: 4x + y = 1000 and we KNow : Area per pen = xy/3
anonymous
  • anonymous
yea perfect
amistre64
  • amistre64
yes, since it is <0 it is a max
amistre64
  • amistre64
we want max area per pen so we substitute y in our area formula and get the derivative.
amistre64
  • amistre64
when y' = 0 we have a bend, a max OR a min.
anonymous
  • anonymous
which we just say is 1000-8x
amistre64
  • 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
  • anonymous
so y''= -8
amistre64
  • amistre64
tells us it is always concave DOWN becasue its always (-8)
amistre64
  • amistre64
you still confused? or feeling more confident?
anonymous
  • anonymous
yea im getting it a little. He said since x fits in the open interval, you take the 2nd derivative
anonymous
  • anonymous
ahhhh where do I plug in our x in our second derivative?
amistre64
  • 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
  • anonymous
yea
amistre64
  • amistre64
x^3 changes....so that youd check :) but we aint messin with no x^3
anonymous
  • anonymous
what is A'' of (1000x-4x^2)/3
anonymous
  • anonymous
lets see if it works anyway
amistre64
  • amistre64
A' = (1000-8x)/3 A'' = -8/3 A''' = 0
anonymous
  • anonymous
thats what i thought
amistre64
  • amistre64
you say this is calc 2 stuff?
anonymous
  • anonymous
yea why?
amistre64
  • amistre64
cause im only in trig at the moment.....and it is sooooo boring there.
anonymous
  • anonymous
haha wow
anonymous
  • anonymous
how are you helping me so well then?
amistre64
  • amistre64
i dunno, maybe I is smart ? I taught myself calculus in intermediate algebra to pass the time....
anonymous
  • anonymous
haha nice yea this is college calc 2
amistre64
  • amistre64
I still need alot of practice with integrals tho..... and word problems. I loathe word problems :)
anonymous
  • anonymous
yea you are still decent
amistre64
  • amistre64
Im gonna get my masters in math...and teach college math
amistre64
  • amistre64
:) thanx
anonymous
  • anonymous
i bet you can do it! easy 4 u
amistre64
  • amistre64
easy as....should I?......PI lol
anonymous
  • anonymous
lol
anonymous
  • anonymous
thx so much though
amistre64
  • amistre64
no prob... keep up the good work :)
anonymous
  • anonymous
my friend confirmed everything
anonymous
  • anonymous
its right

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