Question

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.

Answer 1

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

Answer 2

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

Answer 3

you dont need fence for the river part

Answer 4

so no

Answer 5

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

Answer 6

yea the 2 equations i set up are my up top

Answer 7

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

Answer 8

no prob

Answer 9

take your time

Answer 10

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

Answer 11

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

Answer 12

yea perfect

Answer 13

the deriviate and solve for max right?

Answer 14

esxucse typos, my keyboard hates me:)

Answer 15

no prob, yea

Answer 16

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

Answer 17

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

Answer 18

x = 3000/24 = 1500/12 = 750/6.....

yeah substitute in y=1000-4x.

Answer 19

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

Answer 20

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

Answer 21

sorry im going slow

Answer 22

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

Answer 23

was my A(x) correct?

Answer 24

Your A(x) was correct...

Answer 25

then you took the derivative?

Answer 26

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

Answer 27

You remember your quotient rule?

Answer 28

yea let me do it one sec

Answer 29

3000-24x all over 9

Answer 30

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

Answer 31

y did i get x=27000/24

Answer 32

solve for y' = 0.

What makes (3000-24x)/9 = 0?

Answer 33

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

Answer 34

3000-24x = 0 what is x?

Answer 35

so 1000-8x

Answer 36

thats good too

Answer 37

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

Answer 38

1000/8=x

Answer 39

125=x

Answer 40

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

Good job

Answer 41

y=1000-4(125)

Answer 42

yep....

Answer 43

A=1/3(500)*125

Answer 44

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

Answer 45

20833ft^2, that seems large?

Answer 46

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

Answer 47

oh ok , now the other part is the maximum

Answer 48

2nd derivative right?

Answer 49

total inclosed area is 62500 ft^2

Answer 50

20833 1/3 each

Answer 51

they just want individual

Answer 52

so 20833ft^2 per pen

Answer 53

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

Answer 54

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

Answer 55

i need to do all steps she takes off for everything

Answer 56

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

Answer 57

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

Answer 58

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

Answer 59

k haha

Answer 60

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

Answer 61

soundds good

Answer 62

it should be negative to be a max
<0

Answer 63

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

Answer 64

i mean in concavity relation

Answer 65

if its >0, its concave up oh okay

Answer 66

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

Answer 67

how do you have a max then

Answer 68

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

Answer 69

oh okay

Answer 70

hold up is the second derivative -8

Answer 71

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

Answer 72

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

Answer 73

oh i see

Answer 74

sorry, lol i am so bad at optimization

Answer 75

:)

Answer 76

how do the domains work

Answer 77

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

Answer 78

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

Answer 79

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

Answer 80

so (0, Infinity)

Answer 81

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.

Answer 82

wait....just had athought

Answer 83

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

Answer 84

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 :)

Answer 85

she likes simplicity

Answer 86

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

Answer 87

my friend said the domain is 0 to 250

Answer 88

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

Answer 89

its your x

Answer 90

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

Answer 91

and thats your A( )>0

Answer 92

which is your max by the Extreme value theorem

Answer 93

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

Answer 94

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

Answer 95

thats what the 250 is

Answer 96

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

Answer 97

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

Answer 98

is 125 okay because it falls in the domain?