What is the Growth Constant and World Population at Any Time?

[brad sue]

Hi, I don't know what I am doing wrong in this problem:

In 1980 the world population was approximately 4.5 billion and in the year 2000 it was approximately 6 billion. Assume that the world population at each time t increases at a rate proprtional to the population at time t. Measure t in year after 1980.
Find the growth constant and give the world population at any time t.

What I did is I set the year 1980 as reference.A(t=0)= A_o=4.5
A(20)=6.

we know that A(t)=A_o*e^r*t. (r =rate)
so I get 6=A(t)=4.5*e^r*20.
then I solve for r. I found r=.014.
The solution the teacher not is r=.0231

Can someone please tell me what I missed?
Thank you
B

 

[mathwonk]

forgive me for not answering your question. 9i am sure you can answer it yourself witha calculator.

the main thing wrong with yo0ur problem is the incorrect aassumption that population satisfiesd the de P' = cP. this is =easily discredited by looking at population figures for the last 200 years or so.


more accxurate is the "logistic" equation P' = cP(1 - P/N).


in this model the population levels off as t goes to infinity, obviously more believable since the Earth is finite. with your model the people of the Earth would be literally standing on each others shoulders in a few hundred years.

 

[HallsofIvy]

Mathwonk, while everything you say is true, it won't help brad sue!

The problem as stated said " Assume that the world population at each time t increases at a rate proprtional to the population at time t."- a reasonable simplification of what really happens.

brad sue, assuming the problem really is as you stated, then r= 0.014 is approximately correct. You might want to ask your teacher how he/she got that 0.0231.