Population Problem: Solve for When 51003 in Brampton

[fomenkoa]

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Hi everyone

I'm having trouble with this problem, while preparing for my calculus test this Monday...could anyone help me figure out the solution...I know that I must use the formula P=P(original)e^kt where t is time, and k is the growth factor but I'm not sure how to work with the years:

Over a span of 20 years the population of Bramptom increased from 51003 to 149030. The population of Brampton in 2000 was 310792.

a)Assuming the population model applies to the entire relevant domain when was the population 51003?


Thanks a lot
Anton

 

[arildno]

It is simplest to let t=0 correspond to year 2000.
Hence, you may write:
$$P(t)=(310792)e^{kt}$$

You have two unknowns here:
k
and t*, the time measured from t=0 when the population is 51003 (the other time is, of course, t*+20)

t* will be negative.

 

[fomenkoa]

I still can't figure it out...all these years and variables are confusing

Is

49030 = 51003e^k(t+20) right as a first formula?

I'm really confused :tongue2:

 

[quasar987]

Is 1966 the answer?

 

[fomenkoa]

Well, yeah but I can't figure out the method of solution...which two fomulas to create to determine k

 

[arildno]

Now, can we agree on one thing:

It does not really matter which instant we choose to assign the time-value "0",

but once we've made our choice, we should proceed in a CONSISTENT manner?

 

[fomenkoa]

Yeah...but how can one integrate a random period of 20 years with no dates given and the year 2000...that's what gets me confused...we don't know the dates when the 20-year period takes place

 

[quasar987]

The way I did it is by first setting t = 0 at the time where the population was 51003. Then, P(20) = 149030 = 51003 exp(20k) and this allowed me to find the value of k.

Then, by setting t = 0 at birth of Christ, I found P(0) by setting P(2000) = 310792 and solving for P(0), which gave the dubious "something^(-42)" :grumpy: .. Then, I just isolated ? in P(?) = 51003 = P(0) exp(?t).

Does this make sense to you? The key is that k has the same value no matter where you set t = 0 because the growing rate of the population is the same at any moment of its history.

 

[arildno]

Yeah...but how can one integrate a random period of 20 years with no dates given and the year 2000...that's what gets me confused...we don't know the dates when the 20-year period takes place

It is precisely for this reason it is easiest to choose our known year as t=0.

As I've shown, this means as a function of time, we have:

$$P(t)=310792e^{kt}$$

Note that P(0)=310792, which is the population in year "2000", which we chose to correspond to t=0.

Your two equations are now:
P(t*)=51003
P(t*+20)=149030

These equations can now be solved for t* and k.