Question

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?

Answer 1

have any ideas wat to do? :)

Answer 2

or how to start off?

Answer 3

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

Answer 4

no need for one

Answer 5

but it's helpful

Answer 6

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

Answer 7

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

Answer 8

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

Answer 9

then you plug in 30 for t

Answer 10

wat is the denominator under t?

Answer 11

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

Answer 12

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

Answer 13

D: haha

Answer 14

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

Answer 15

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

Answer 16

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

Answer 17

where does 1.15 come from?

Answer 18

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

Answer 19

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

Answer 20

but the question says 15% :/

Answer 21

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.

Answer 22

lol forget me, answer the girl :P

Answer 23

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

Answer 24

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.

Answer 25

okay..

Answer 26

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

Answer 27

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

Answer 28

o.o where is that from?

Answer 29

try this http://qrc.depaul.edu/StudyGuide2009/Notes/Savings%20Accounts/Compound%20Interest.htm

Answer 30

its the formula given

Answer 31

what math class are you in?

Answer 32

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

Answer 33

from that formula we integrate and so on

Answer 34

wow, you've confused me.

Answer 35

XD

Answer 36

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

Answer 37

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

Answer 38

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

Answer 39

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

Answer 40

what does A represent?

Answer 41

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

Answer 42

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

Answer 43

http://fym.la.asu.edu/~pvaz/mat272/ch_07.pdf

Answer 44

wats the answer?

Answer 45

7604 people

Answer 46

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

Answer 47

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

Answer 48

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

Answer 49

okay i'll try

Answer 50

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

Answer 51

do i have to find a new rate or what?

Answer 52

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.

Answer 53

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?

Answer 54

i got it! thanks guys for the help :)

Answer 55

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

Answer 56

but you guys made it more clear so thanks :D