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shaqadry

The population of a town grows at a rate proportional to the population present at time t. The initial population of 5000 people is increased by 15% in the period of 10 years. What will the estimated population be in 30 years time?

  • one year ago
  • one year ago

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  1. HawkCrimson
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    have any ideas wat to do? :)

    • one year ago
  2. HawkCrimson
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    or how to start off?

    • one year ago
  3. Peter14
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    do you know an equation that works for this kind of thing?

    • one year ago
  4. HawkCrimson
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    no need for one

    • one year ago
  5. Peter14
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    but it's helpful

    • one year ago
  6. shaqadry
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    yes. i work it out until i got P = Ae^(0.15t)

    • one year ago
  7. HawkCrimson
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    since 10yrs=15% then 30yrs=45% thus the population growth in 30 yrs is 5000 x 145/100 50 x 145

    • one year ago
  8. Peter14
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    the equation you can find for this problem is\[P=5000 \times 1.15^\frac{ t }{ 10 }\]

    • one year ago
  9. Peter14
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    then you plug in 30 for t

    • one year ago
  10. HawkCrimson
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    wat is the denominator under t?

    • one year ago
  11. HawkCrimson
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    lol the shortcut is easier and less complicated :P ur just jelly

    • one year ago
  12. Peter14
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    nevermind, @HawkCrimson the shortcut you took doesn't work

    • one year ago
  13. HawkCrimson
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    D: haha

    • one year ago
  14. shaqadry
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    @Peter14 i dont understand...can you show me how it starts from P = Ae^(0.15t) ?

    • one year ago
  15. Peter14
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    what you actually do each 10 years is you multiply the number of people by 1.15

    • one year ago
  16. HawkCrimson
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    cause it's 15%= . 15 right? :D

    • one year ago
  17. shaqadry
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    where does 1.15 come from?

    • one year ago
  18. HawkCrimson
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    oh ik how to do it by the shortcut way :P but it's long and tedius

    • one year ago
  19. Peter14
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    1.15 is 115% the way the problem expresses this is as an exponential growth problem.

    • one year ago
  20. shaqadry
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    but the question says 15% :/

    • one year ago
  21. Peter14
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    the shortcut you took would work for an arithmetic sequence (meaning you add a certain amount each time) but this is a geometric sequence (meaning you multiply by a certain amount each time.

    • one year ago
  22. HawkCrimson
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    lol forget me, answer the girl :P

    • one year ago
  23. HawkCrimson
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    100%=5000 pop so if it has a pop increase by 15% then it will become 115% that is y

    • one year ago
  24. Peter14
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    the question says the population increases by 15% each time. The number you get shouldn't decrease at all. Let's look at the first case. If you multiply 5000 by 15% you get the increase in population: 750 people. You can then add that to 5000 to get 5750. or on the other hand, you can multiply 5000 by 115% (because 115% is 100%+15%) and that gives you 5750 with only one operation.

    • one year ago
  25. shaqadry
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    okay..

    • one year ago
  26. Peter14
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    have you looked at compound interest in your math class or should I try to explain that to you?

    • one year ago
  27. shaqadry
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    how do i calculate using the formula dP/dt = kP ?

    • one year ago
  28. Peter14
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    o.o where is that from?

    • one year ago
  29. Peter14
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    try this http://qrc.depaul.edu/StudyGuide2009/Notes/Savings%20Accounts/Compound%20Interest.htm

    • one year ago
  30. shaqadry
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    its the formula given

    • one year ago
  31. Peter14
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    what math class are you in?

    • one year ago
  32. shaqadry
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    i mean thats what we study in application of first order differential equations

    • one year ago
  33. shaqadry
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    from that formula we integrate and so on

    • one year ago
  34. Peter14
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    wow, you've confused me.

    • one year ago
  35. HawkCrimson
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    XD

    • one year ago
  36. Peter14
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    and google is sooo slow here in china so I have to stay confused...

    • one year ago
  37. Peter14
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    ok, I understand now. you were given dP/dt = kP as the equation at the beginning. I get it.

    • one year ago
  38. Peter14
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    what was the exact wording of the problem? I seem to have misunderstood it before.

    • one year ago
  39. shaqadry
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    i just want to know how do i start from P=Ae^(0.15t) from what i integrated from the above formula

    • one year ago
  40. Peter14
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    what does A represent?

    • one year ago
  41. Peter14
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    I think T would be 3, because it seems from the description at the top that T is in intervals of 10 years and then P looks like your final population

    • one year ago
  42. shaqadry
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    yeaa tried that but couldnt seem to get the answer right :/

    • one year ago
  43. shaqadry
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    http://fym.la.asu.edu/~pvaz/mat272/ch_07.pdf

    • one year ago
  44. HawkCrimson
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    wats the answer?

    • one year ago
  45. shaqadry
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    7604 people

    • one year ago
  46. Peter14
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    you get that using the formula for compound interest, but I think they want you to show your work from using the methods they give you

    • one year ago
  47. HawkCrimson
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    i got that using the shortcut method but that wont get u any marks in the exam, ill try using the formulae

    • one year ago
  48. Peter14
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    maybe you need to find the annual rate of increase instead of the rate of increase for 10 years

    • one year ago
  49. shaqadry
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    okay i'll try

    • one year ago
  50. HawkCrimson
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    P = Ae^0.15t P = 5000e^0.15(3) P = 7841 :/

    • one year ago
  51. shaqadry
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    do i have to find a new rate or what?

    • one year ago
  52. Peter14
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    yes, you do and I know by doing some graphing tricks that the rate ends up approximately 0.013975, but I don't know exactly how you get that numerically. It should have something to do with ln something.

    • one year ago
  53. Peter14
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    so in the textbook chapter you posted the final equation is P=A(e^kt) you are trying to find k given t=30 and a=5000 do they give you P? I guess what i'm trying to say is, what exactly do they give you at the start of the problem?

    • one year ago
  54. shaqadry
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    i got it! thanks guys for the help :)

    • one year ago
  55. Peter14
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    I think you got it less because of us and more despite us.

    • one year ago
  56. shaqadry
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    but you guys made it more clear so thanks :D

    • one year ago
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