## shaqadry 2 years ago 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?

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1. HawkCrimson

have any ideas wat to do? :)

2. HawkCrimson

or how to start off?

3. Peter14

do you know an equation that works for this kind of thing?

4. HawkCrimson

no need for one

5. Peter14

yes. i work it out until i got P = Ae^(0.15t)

7. HawkCrimson

since 10yrs=15% then 30yrs=45% thus the population growth in 30 yrs is 5000 x 145/100 50 x 145

8. Peter14

the equation you can find for this problem is$P=5000 \times 1.15^\frac{ t }{ 10 }$

9. Peter14

then you plug in 30 for t

10. HawkCrimson

wat is the denominator under t?

11. HawkCrimson

lol the shortcut is easier and less complicated :P ur just jelly

12. Peter14

nevermind, @HawkCrimson the shortcut you took doesn't work

13. HawkCrimson

D: haha

@Peter14 i dont understand...can you show me how it starts from P = Ae^(0.15t) ?

15. Peter14

what you actually do each 10 years is you multiply the number of people by 1.15

16. HawkCrimson

cause it's 15%= . 15 right? :D

where does 1.15 come from?

18. HawkCrimson

oh ik how to do it by the shortcut way :P but it's long and tedius

19. Peter14

1.15 is 115% the way the problem expresses this is as an exponential growth problem.

but the question says 15% :/

21. Peter14

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.

22. HawkCrimson

lol forget me, answer the girl :P

23. HawkCrimson

100%=5000 pop so if it has a pop increase by 15% then it will become 115% that is y

24. Peter14

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.

okay..

26. Peter14

have you looked at compound interest in your math class or should I try to explain that to you?

how do i calculate using the formula dP/dt = kP ?

28. Peter14

o.o where is that from?

29. Peter14

its the formula given

31. Peter14

what math class are you in?

i mean thats what we study in application of first order differential equations

from that formula we integrate and so on

34. Peter14

wow, you've confused me.

35. HawkCrimson

XD

36. Peter14

and google is sooo slow here in china so I have to stay confused...

37. Peter14

ok, I understand now. you were given dP/dt = kP as the equation at the beginning. I get it.

38. Peter14

what was the exact wording of the problem? I seem to have misunderstood it before.

i just want to know how do i start from P=Ae^(0.15t) from what i integrated from the above formula

40. Peter14

what does A represent?

41. Peter14

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

yeaa tried that but couldnt seem to get the answer right :/

44. HawkCrimson

7604 people

46. Peter14

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

47. HawkCrimson

i got that using the shortcut method but that wont get u any marks in the exam, ill try using the formulae

48. Peter14

maybe you need to find the annual rate of increase instead of the rate of increase for 10 years

okay i'll try

50. HawkCrimson

P = Ae^0.15t P = 5000e^0.15(3) P = 7841 :/

do i have to find a new rate or what?

52. Peter14

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.

53. Peter14

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?

i got it! thanks guys for the help :)

55. Peter14

I think you got it less because of us and more despite us.