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anonymous

  • 5 years ago

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.

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  1. anonymous
    • 5 years ago
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    So i got that the equation will be F=1000ft=4x+y and A=1/3xy

  2. amistre64
    • 5 years ago
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    Is the river included as one side of the pens? cause in alot of problems it is..

  3. anonymous
    • 5 years ago
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    you dont need fence for the river part

  4. anonymous
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    so no

  5. amistre64
    • 5 years ago
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    then we need 4x + y = 1000, right?

  6. anonymous
    • 5 years ago
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    yea the 2 equations i set up are my up top

  7. amistre64
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    good, but I need to work it thru in my head too:)

  8. anonymous
    • 5 years ago
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    no prob

  9. anonymous
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    take your time

  10. amistre64
    • 5 years ago
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    and the area of each pen is x(y/3)

  11. amistre64
    • 5 years ago
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    Since y = 1000 - 4x, can we use that in the area?

  12. anonymous
    • 5 years ago
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    yea perfect

  13. amistre64
    • 5 years ago
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    the deriviate and solve for max right?

  14. amistre64
    • 5 years ago
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    esxucse typos, my keyboard hates me:)

  15. anonymous
    • 5 years ago
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    no prob, yea

  16. amistre64
    • 5 years ago
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    i get (3000 - 24x)/9 for a derivative...what about you?

  17. anonymous
    • 5 years ago
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    hold on, did you set it up first by substituting?

  18. amistre64
    • 5 years ago
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    x = 3000/24 = 1500/12 = 750/6..... yeah substitute in y=1000-4x.

  19. anonymous
    • 5 years ago
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    so A(x)=1000x-4X^2 all over 3

  20. amistre64
    • 5 years ago
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    x = 375/3...x = 125 for me

  21. anonymous
    • 5 years ago
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    sorry im going slow

  22. amistre64
    • 5 years ago
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    its ok...gives me time for my mind to wander :)

  23. anonymous
    • 5 years ago
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    was my A(x) correct?

  24. amistre64
    • 5 years ago
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    Your A(x) was correct...

  25. anonymous
    • 5 years ago
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    then you took the derivative?

  26. amistre64
    • 5 years ago
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    yep, the derivative tells us where we can find bends inthe graph that are our max and min values.

  27. amistre64
    • 5 years ago
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    You remember your quotient rule?

  28. anonymous
    • 5 years ago
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    yea let me do it one sec

  29. anonymous
    • 5 years ago
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    3000-24x all over 9

  30. amistre64
    • 5 years ago
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    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

  31. anonymous
    • 5 years ago
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    y did i get x=27000/24

  32. amistre64
    • 5 years ago
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    solve for y' = 0. What makes (3000-24x)/9 = 0?

  33. amistre64
    • 5 years ago
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    the top has to equal 0 for ay of it to matter; so throw away the 9

  34. amistre64
    • 5 years ago
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    3000-24x = 0 what is x?

  35. anonymous
    • 5 years ago
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    so 1000-8x

  36. amistre64
    • 5 years ago
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    thats good too

  37. amistre64
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    1000 - 8x = 0...is the same thing

  38. anonymous
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    1000/8=x

  39. anonymous
    • 5 years ago
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    125=x

  40. amistre64
    • 5 years ago
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    yep, now reduce it or use it as is if you want :) Good job

  41. anonymous
    • 5 years ago
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    y=1000-4(125)

  42. amistre64
    • 5 years ago
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    yep....

  43. anonymous
    • 5 years ago
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    A=1/3(500)*125

  44. amistre64
    • 5 years ago
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    (500)(125)/3 is more accurate, but yeah...

  45. anonymous
    • 5 years ago
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    20833ft^2, that seems large?

  46. amistre64
    • 5 years ago
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    when talking about football field measurements, I imagine it would be :)

  47. anonymous
    • 5 years ago
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    oh ok , now the other part is the maximum

  48. anonymous
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    2nd derivative right?

  49. amistre64
    • 5 years ago
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    total inclosed area is 62500 ft^2

  50. amistre64
    • 5 years ago
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    20833 1/3 each

  51. anonymous
    • 5 years ago
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    they just want individual

  52. anonymous
    • 5 years ago
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    so 20833ft^2 per pen

  53. amistre64
    • 5 years ago
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    we already found the maximum... there is only one hump in a quadratic..

  54. amistre64
    • 5 years ago
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    you can second derive it if you want, just to prove it to yourself, but it aint needed

  55. anonymous
    • 5 years ago
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    i need to do all steps she takes off for everything

  56. anonymous
    • 5 years ago
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    Is this 100% right, she had like inequality equations and domain stuff on her examples. I am confused

  57. amistre64
    • 5 years ago
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    lol..when y'' is (+) we have a concave down which is an upside down bowl on the graph. And that is the max.

  58. amistre64
    • 5 years ago
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    its right....so far :) use the math luke.....the maaatthhhh...

  59. anonymous
    • 5 years ago
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    k haha

  60. amistre64
    • 5 years ago
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    might wanna double check that concavity thing, I never use it enough to remember if Im right :)

  61. anonymous
    • 5 years ago
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    soundds good

  62. anonymous
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    it should be negative to be a max <0

  63. amistre64
    • 5 years ago
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    remember CAVE....CAVE UP is (+) and CAVE DOWN is (-) or the other way around..

  64. anonymous
    • 5 years ago
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    i mean in concavity relation

  65. anonymous
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    if its >0, its concave up oh okay

  66. amistre64
    • 5 years ago
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    lol....ill trust you on that :)

  67. anonymous
    • 5 years ago
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    how do you have a max then

  68. amistre64
    • 5 years ago
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    picture the graph of (-x^2) thats all this is.

  69. anonymous
    • 5 years ago
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    oh okay

  70. anonymous
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    hold up is the second derivative -8

  71. amistre64
    • 5 years ago
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    thats right, so the cave is down. its an upside down bowl.

  72. amistre64
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    -(x^2) ... y'=-(2x)....y'' = -(2)

  73. anonymous
    • 5 years ago
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    oh i see

  74. anonymous
    • 5 years ago
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    sorry, lol i am so bad at optimization

  75. amistre64
    • 5 years ago
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    :)

  76. anonymous
    • 5 years ago
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    how do the domains work

  77. anonymous
    • 5 years ago
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    she wants us to show that too D:(0, )

  78. amistre64
    • 5 years ago
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    there is nothing limiting our domain, so it canbe any real number (R)

  79. amistre64
    • 5 years ago
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    look at the fractions we made, if anything makes the bottom a 0, throw it out. Nothing matters, so keep it all

  80. anonymous
    • 5 years ago
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    so (0, Infinity)

  81. amistre64
    • 5 years ago
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    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.

  82. amistre64
    • 5 years ago
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    wait....just had athought

  83. amistre64
    • 5 years ago
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    we only have so much fencing to work with... so make the D [0,1000]

  84. amistre64
    • 5 years ago
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    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 :)

  85. anonymous
    • 5 years ago
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    she likes simplicity

  86. amistre64
    • 5 years ago
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    my gut tells me that we do [0,1000]..

  87. anonymous
    • 5 years ago
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    my friend said the domain is 0 to 250

  88. amistre64
    • 5 years ago
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    does your friend tell you how to find the max area :)

  89. anonymous
    • 5 years ago
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    its your x

  90. anonymous
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    He said solve 1000x-4x^2 for x

  91. anonymous
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    and thats your A( )>0

  92. anonymous
    • 5 years ago
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    which is your max by the Extreme value theorem

  93. amistre64
    • 5 years ago
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    thats fine.....well use 0 to 250......its as good as anyother I can think of :)

  94. anonymous
    • 5 years ago
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    no he said solve the inequality 1000-4x>0

  95. anonymous
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    thats what the 250 is

  96. amistre64
    • 5 years ago
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    1000/4 = 250....i know iknow...yeahyeahyeah....

  97. anonymous
    • 5 years ago
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    can we walk through it step by step, I am so confused

  98. anonymous
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    is 125 okay because it falls in the domain?

  99. amistre64
    • 5 years ago
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    125 is good

  100. amistre64
    • 5 years ago
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    step 1: we are given certian rules that govern our equations...did we use them right?

  101. anonymous
    • 5 years ago
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    yes

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

  103. amistre64
    • 5 years ago
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    we Know: 4x + y = 1000 and we KNow : Area per pen = xy/3

  104. anonymous
    • 5 years ago
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    yea perfect

  105. amistre64
    • 5 years ago
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    yes, since it is <0 it is a max

  106. amistre64
    • 5 years ago
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    we want max area per pen so we substitute y in our area formula and get the derivative.

  107. amistre64
    • 5 years ago
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    when y' = 0 we have a bend, a max OR a min.

  108. anonymous
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    which we just say is 1000-8x

  109. amistre64
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    yes.... and y'' tells us how the graph behaves at certain points. (-) = CAVE DOWN = max point (+) = CAVE UP = min point (0) = point of inflection.

  110. anonymous
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    so y''= -8

  111. amistre64
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    tells us it is always concave DOWN becasue its always (-8)

  112. amistre64
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    you still confused? or feeling more confident?

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

  114. anonymous
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    ahhhh where do I plug in our x in our second derivative?

  115. amistre64
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    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

  116. anonymous
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    yea

  117. amistre64
    • 5 years ago
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    x^3 changes....so that youd check :) but we aint messin with no x^3

  118. anonymous
    • 5 years ago
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    what is A'' of (1000x-4x^2)/3

  119. anonymous
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    lets see if it works anyway

  120. amistre64
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    A' = (1000-8x)/3 A'' = -8/3 A''' = 0

  121. anonymous
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    thats what i thought

  122. amistre64
    • 5 years ago
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    you say this is calc 2 stuff?

  123. anonymous
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    yea why?

  124. amistre64
    • 5 years ago
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    cause im only in trig at the moment.....and it is sooooo boring there.

  125. anonymous
    • 5 years ago
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    haha wow

  126. anonymous
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    how are you helping me so well then?

  127. amistre64
    • 5 years ago
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    i dunno, maybe I is smart ? I taught myself calculus in intermediate algebra to pass the time....

  128. anonymous
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    haha nice yea this is college calc 2

  129. amistre64
    • 5 years ago
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    I still need alot of practice with integrals tho..... and word problems. I loathe word problems :)

  130. anonymous
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    yea you are still decent

  131. amistre64
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    Im gonna get my masters in math...and teach college math

  132. amistre64
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    :) thanx

  133. anonymous
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    i bet you can do it! easy 4 u

  134. amistre64
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    easy as....should I?......PI lol

  135. anonymous
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    lol

  136. anonymous
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    thx so much though

  137. amistre64
    • 5 years ago
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    no prob... keep up the good work :)

  138. anonymous
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    my friend confirmed everything

  139. anonymous
    • 5 years ago
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    its right

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