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anonymous

  • 5 years ago

Need help with difference equation...

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  1. anonymous
    • 5 years ago
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    Suppose your car has a 14 gallon gas tank that you fill as soon as the level drops below half a tank. Every time you fill up, you add 1 quart (1/4 gallon) of an additive that mixes completely with the gas and is used along with the gas.

  2. anonymous
    • 5 years ago
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    What is the difference equation and initial value that models the amount A sub n of additive in the tank at the time of the nth fill-up?

  3. amistre64
    • 5 years ago
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    found it :)

  4. anonymous
    • 5 years ago
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    great!

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  5. anonymous
    • 5 years ago
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    I attached so I will not have to type out entire problems

  6. amistre64
    • 5 years ago
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    i see it. how much additive is remaining at the time of each fill up..

  7. anonymous
    • 5 years ago
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    I truly appreciate your help...I love getting on here helping others so I share your passion!

  8. amistre64
    • 5 years ago
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    pistachios are more of my passion i think :)

  9. amistre64
    • 5 years ago
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    this is more of a hobby lol

  10. anonymous
    • 5 years ago
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    lol...well you're AWESOME!

  11. anonymous
    • 5 years ago
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    ok so do you understand the question

  12. amistre64
    • 5 years ago
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    lets assume that we start with a clean tank..right? at halhway we fill it up and add the additive.

  13. anonymous
    • 5 years ago
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    ok

  14. amistre64
    • 5 years ago
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    at half a tank we have only half of what we put still in..... and the ration is the same, 14: .25 or some such right?

  15. anonymous
    • 5 years ago
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    yes

  16. amistre64
    • 5 years ago
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    the quart we put in becomes half a quart, and we add more gas and anouter quart... right?

  17. anonymous
    • 5 years ago
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    ok

  18. amistre64
    • 5 years ago
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    this loks to be one of those exponential or log setups... from first glance

  19. anonymous
    • 5 years ago
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    don't think so

  20. amistre64
    • 5 years ago
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    I could be wrong, I allow that :)

  21. amistre64
    • 5 years ago
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    So A0 = 1 quart of additive. A1 = 1 quart + 1/2 thats left = 1.5 quarts a2 = 1.5/2 +1 and iterates that way...makes sense?

  22. anonymous
    • 5 years ago
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    yes!

  23. amistre64
    • 5 years ago
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    lets plug in the numbers and see if theres a pattern to follow :)

  24. anonymous
    • 5 years ago
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    would it be easier to stay with gallons since the tank is in gallons A1=.25+7?

  25. amistre64
    • 5 years ago
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    "that models the amount An of additive in the tank at the time" leads me to believe that the gas itself is of no consquence, only to tell us when to fill up again

  26. anonymous
    • 5 years ago
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    ok

  27. amistre64
    • 5 years ago
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    1 = 1 2 = 1.5 3 = 1.75 4 = 1.875 5 = 1.9375 6 = 1.96875

  28. amistre64
    • 5 years ago
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    7 = 1.984375

  29. amistre64
    • 5 years ago
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    i see a pattern emerging, just gotta put my finger on it :)

  30. anonymous
    • 5 years ago
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    I got it An=.5A0+1?

  31. amistre64
    • 5 years ago
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    \[\frac{\frac{\frac{\frac{1}{2}+1}{2}+1}{2}+1}{2}+1...\]

  32. amistre64
    • 5 years ago
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    Does this pattern remind you of anything?

  33. anonymous
    • 5 years ago
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    Fibonacci?

  34. amistre64
    • 5 years ago
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    it does kinda look fibonaci like :)

  35. amistre64
    • 5 years ago
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    i cant place it yet....

  36. amistre64
    • 5 years ago
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    but its growing to a limit of 1.99999999999999999999

  37. anonymous
    • 5 years ago
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    Can you help with others/check

  38. anonymous
    • 5 years ago
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    yes I have 1.998 for n=10

  39. amistre64
    • 5 years ago
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    I can try; have you finished them?

  40. anonymous
    • 5 years ago
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    Still have to answer rest of 2, almost done with 3, 4 complete-easy

  41. anonymous
    • 5 years ago
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    the formula on p.368 is \[D _{n}=L-(L-D _{0})a ^{n}\]

  42. amistre64
    • 5 years ago
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    and what do we do with this formula :)

  43. anonymous
    • 5 years ago
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    c) Find the solution from a using equation

  44. anonymous
    • 5 years ago
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    or maybe we use L =B/1-a as n approaches infinity

  45. amistre64
    • 5 years ago
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    have we found our difference equation yet?

  46. anonymous
    • 5 years ago
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    I am using Asub n = .5Asub0 + 1

  47. anonymous
    • 5 years ago
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    typed it earlier...do you agree?

  48. amistre64
    • 5 years ago
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    is that the same as .05(n) + 1?

  49. amistre64
    • 5 years ago
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    .5(n) +1

  50. anonymous
    • 5 years ago
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    yes!

  51. amistre64
    • 5 years ago
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    at n = 3 we get: 2.5 with that one, so it doesnt match our data.

  52. amistre64
    • 5 years ago
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    at any rate, we are adding 1 and a fraction, so anything over 2 is not correct :)

  53. anonymous
    • 5 years ago
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    using previous values of Asubn to get others

  54. anonymous
    • 5 years ago
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    \[A _{n}=.5A _{n+1}+1\]

  55. amistre64
    • 5 years ago
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    lets test that equation and see if it gets us what we know. .5(3+1) + 1 = 2+1 = 3. its not working yet :)

  56. amistre64
    • 5 years ago
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    I got something :) needs to be ironed out tho

  57. anonymous
    • 5 years ago
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    \[A _{n+1}=.5A _{n}+1\]

  58. amistre64
    • 5 years ago
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    2n+1 An = 1 + ------ 2^n

  59. anonymous
    • 5 years ago
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    I see it but can't explain in formula correctly

  60. amistre64
    • 5 years ago
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    \[1,\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, \frac{31}{32}, \frac{63}{64}\]

  61. amistre64
    • 5 years ago
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    that fits it to a "T" :)

  62. amistre64
    • 5 years ago
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    \[A _{n} = 1+\frac{2n+1}{2^n}\]

  63. amistre64
    • 5 years ago
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    its off at n=0; do we start at 0?

  64. anonymous
    • 5 years ago
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    that is giving greater than 2

  65. anonymous
    • 5 years ago
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    no n=1

  66. amistre64
    • 5 years ago
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    its close, but needs to be tweeked

  67. amistre64
    • 5 years ago
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    2(n-1)+1

  68. amistre64
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    \[A_{n} = 1 + \frac{2(n-1)+1}{2^n}\]

  69. amistre64
    • 5 years ago
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    not yet eh lol

  70. anonymous
    • 5 years ago
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    \[A _{n}=.5A _{n-1}+1\]

  71. anonymous
    • 5 years ago
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    we are taking half of previous output and adding 1 right?

  72. amistre64
    • 5 years ago
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    yes

  73. amistre64
    • 5 years ago
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    yours looks good like that :)

  74. amistre64
    • 5 years ago
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    half of what it was, then add 1....yep

  75. amistre64
    • 5 years ago
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    What is A_0-1? or are we starting n=1 :)

  76. anonymous
    • 5 years ago
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    geeshhh. I had that thinking too hard ; (

  77. amistre64
    • 5 years ago
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    lol

  78. anonymous
    • 5 years ago
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    n=1

  79. amistre64
    • 5 years ago
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    good, we got an equation that we like...now we use it in C right? or B first

  80. anonymous
    • 5 years ago
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    a

  81. amistre64
    • 5 years ago
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    that was A; find the equation ;)

  82. amistre64
    • 5 years ago
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    B is; use it to make a table....

  83. anonymous
    • 5 years ago
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    yes...done

  84. amistre64
    • 5 years ago
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    We just happened to make a table first lol

  85. anonymous
    • 5 years ago
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    working backwards in a problem solving strategy ; )

  86. anonymous
    • 5 years ago
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    is

  87. amistre64
    • 5 years ago
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    Dn =L−(L−D0 )a^n So what does this formula mean? its greek to me :)

  88. anonymous
    • 5 years ago
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    me too...uh oh

  89. amistre64
    • 5 years ago
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    368 should tell us something about it...like what D is and L

  90. amistre64
    • 5 years ago
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    pg 368 that is

  91. anonymous
    • 5 years ago
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    B/1-a in place of L for solutions growing towards infinity

  92. anonymous
    • 5 years ago
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    sorry keep forgetting to his post

  93. amistre64
    • 5 years ago
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    whats the chapter heading for your material?

  94. anonymous
    • 5 years ago
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    Modeling with Difference Equations

  95. amistre64
    • 5 years ago
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    does the formula have a name for it? its probably online somewhere and I can get a better grip on it.

  96. anonymous
    • 5 years ago
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    difference equation...

  97. anonymous
    • 5 years ago
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    using Asubn+1=.5Asubn +1

  98. amistre64
    • 5 years ago
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    we found a first order linear difference equation, if I see it right..

  99. amistre64
    • 5 years ago
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    Tn+1 − Tn = k(Tn − S) looks useful Dn =L−(L−D0 )a^n but I dunno.... are there any other formulas on page 368?

  100. anonymous
    • 5 years ago
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    \[x _{n=}L+(x _{0}-L)a ^{n}\]

  101. anonymous
    • 5 years ago
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    used that in c got b working on e using L=B/1- to find limiting value

  102. anonymous
    • 5 years ago
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    finished d

  103. amistre64
    • 5 years ago
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    you got it?

  104. anonymous
    • 5 years ago
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    need e and f

  105. amistre64
    • 5 years ago
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    the limit is 2 I beleive.... is that right? as n increases without bound we approach 2

  106. anonymous
    • 5 years ago
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    yes but trying to get that from equation...working backwards again

  107. amistre64
    • 5 years ago
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    :) I plugged in alot of iteration in google and gets 2 :)

  108. amistre64
    • 5 years ago
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    are we allowed to use the definition of a limt? f(n+h) - f(n)/h ?

  109. amistre64
    • 5 years ago
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    thats the limit as it approaches zero....not infiinity tho....

  110. amistre64
    • 5 years ago
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    I think we would need a more rigid equation for that than just An=.5A(n-1) + 1 tho, becasue there is no way to determine the value of A(n-1) at any given point.

  111. amistre64
    • 5 years ago
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    unless you got a formula for that too :)

  112. anonymous
    • 5 years ago
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    limit is to infinity ...it also states that n can be replaced by n-1 to convert a forward difference equation to an equivalent backward de

  113. anonymous
    • 5 years ago
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    since n=0

  114. anonymous
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    e) 1/1-.5

  115. amistre64
    • 5 years ago
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    1/.5 = 2

  116. amistre64
    • 5 years ago
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    1/(1-.5) right?

  117. anonymous
    • 5 years ago
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    YEP!!!!

  118. amistre64
    • 5 years ago
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    thats 2 :)

  119. amistre64
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    thats 2 :)

  120. anonymous
    • 5 years ago
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    f)

  121. amistre64
    • 5 years ago
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    .5 would change to a different number

  122. amistre64
    • 5 years ago
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    .6 I believe

  123. amistre64
    • 5 years ago
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    40%full...... .4

  124. anonymous
    • 5 years ago
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    I concur! g)

  125. amistre64
    • 5 years ago
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    .4(An-1) + 1

  126. anonymous
    • 5 years ago
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    forgot post button again

  127. amistre64
    • 5 years ago
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    and for the last one; the size of the tank doesnt matter, its the ratio.

  128. amistre64
    • 5 years ago
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    16 gal filled every half is the same result

  129. anonymous
    • 5 years ago
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    what about limiting value for f

  130. amistre64
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    id venture to guess that is it 2; but let me check my sources :)

  131. amistre64
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    1.666666666666666666666666666......

  132. amistre64
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    if I did it right :)

  133. anonymous
    • 5 years ago
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    just got taht on cal!!!!

  134. anonymous
    • 5 years ago
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    GREAT!

  135. anonymous
    • 5 years ago
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    I have 3a and b can you check c for me Bsub15 = 80000(1.06)to the n - 10,000 so

  136. amistre64
    • 5 years ago
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    \[80000(1.06)n -10000\] looks good

  137. amistre64
    • 5 years ago
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    or is it n-1?

  138. amistre64
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    \[B_n = 80000(1.06)(n-1) - 10000\]

  139. amistre64
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    at 1 that would be a balance of -10000; so prolly not

  140. anonymous
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    \[B _{n+1}=1.06B _{n}-10000\]

  141. anonymous
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    noooo...you were about to make me scream!

  142. amistre64
    • 5 years ago
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    lol

  143. anonymous
    • 5 years ago
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    so c I keep going back and forth on

  144. anonymous
    • 5 years ago
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    got $181,724.66

  145. amistre64
    • 5 years ago
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    well, if he keeps taking 10000 out, he runs outta money before then I think.

  146. amistre64
    • 5 years ago
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    80000(1.06) = 84800 - 10000 = 74800 right?

  147. anonymous
    • 5 years ago
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    yes for n=1

  148. amistre64
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    79288 - 10000 = 69288

  149. amistre64
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    73445.28 - 10000 = 63445.28

  150. amistre64
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    67251.9968 - 10000 = 57251.9968

  151. amistre64
    • 5 years ago
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    60687.116608 - 10000= 50687.116608 53728.34360448 - 10000 = 43728.34 46352.0404 - 10000 = 36352.0404

  152. amistre64
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    were running outta money :)

  153. amistre64
    • 5 years ago
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    8th year: 38533.17 - 10000 = 28533.17 9th year: 30245.16 - 10000 = 20245.16 10th year: 21459.87 - 10000 = 11459.87 11th year: 12147.47 -10000 = 2147.47 12th year: 2276.32 - 10000 ...... cant do it :)

  154. anonymous
    • 5 years ago
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    lol...my 2 year was off by 1,000 wrote wrong digit from cal....geesh

  155. amistre64
    • 5 years ago
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    year 12 shes broke... at 15 theres nothing left;

  156. amistre64
    • 5 years ago
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    if anything she owes 37723.68 by year 15

  157. amistre64
    • 5 years ago
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    so what amount can she withdraw each year that will result in a zero balance at year 15 is the question lol

  158. anonymous
    • 5 years ago
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    yeah i was just reading it again, too lmbo

  159. amistre64
    • 5 years ago
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    its between 8000 and 9000 :)

  160. amistre64
    • 5 years ago
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    between 8200 and 8300 :)

  161. amistre64
    • 5 years ago
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    between 8230 and 8245

  162. anonymous
    • 5 years ago
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    what would be my explanation...you working backwards

  163. amistre64
    • 5 years ago
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    im just reiterating it 15 times on google and narrowing it down :)

  164. amistre64
    • 5 years ago
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    once we get a number, we can plug it in and see if it works :)

  165. amistre64
    • 5 years ago
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    same equation, just a different withdrawal amount

  166. anonymous
    • 5 years ago
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    ok

  167. anonymous
    • 5 years ago
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    would we have to log this

  168. amistre64
    • 5 years ago
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    -8237 = 0.491505273 :)

  169. amistre64
    • 5 years ago
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    - 8 237.03 = -0.206773824 cant get any less - 8 237.02 = 0.0259858751

  170. amistre64
    • 5 years ago
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    explain how you got your answer: trial and error :)

  171. amistre64
    • 5 years ago
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    the way all good people have done it for thousands of years; we narrowed it down til something stuck

  172. anonymous
    • 5 years ago
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    I might add that, lol!

  173. amistre64
    • 5 years ago
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    :)

  174. amistre64
    • 5 years ago
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    plug it into your calculator and see if it gets you a good deal...

  175. anonymous
    • 5 years ago
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    I will finish e bc I know you want to move on...THANKS SOOOO MUCH!

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