1. ## [HELP] Population Growth

i got some problems while i was doing exercises on exponential functions

and here it is. a table set

The question is to make some estimate of time taken for the population to double with a theoretical formula of the form Pt = Po × 2^at
where t is in years and Po and a are some particular values

I reckon that Po is the initial population, but which number should i use?
And how do I find the value of a?
I am really struggling with this topic and would like some help

any help would be highly appreciated
thank you

2. Do you have Excel?. It does a fine job of doing regressions. This is a power regression. You could use an initial population of 9.5 with the year 1960 corresponding to t=0.

Then, 1970 would be t=10 and so forth. For the western population:

Just running it through my TI, I get $y=9.757196\cdot 1.037178^{t}$

In your case, with the 2: $9.757196\cdot 2^{.0526635t}$

Or, in exponential :

$y=9.7572e^{.0365x}$

This one has an R^2 of .9972, which is very accurate. The closer to 1 the better. This is close to 1 for sure.

Their statement of "when will it double?". From where?. The initial population or the latest one?.

Try both. When will it double from the latest of 2*23.9. Plug in $47.8=9.757196\cdot 2^{.0526635t}$ and solve for t.

We get t=43.53. That would about half way through the year 2003.

3. is there a manual way instead of using excel?
like using the numbers and the function itself?

ah i have a TI too
could you guide me how to input the data as well?

4. Just to do a rough estimate. It will not be as accurate as Excel but it'll do OK.

For the western, use 9.5 as the initial population and the last entry of 23.9.

t=25 since it is 25 years from 1960 to 1985.

$23.9=9.5\cdot 2^{25a}$

Solve for a and we get a=.05324044...

That is pretty close to the Excel method, huh?.

Can you solve the above for a?.

Do the same for the developing world one.

what kind of TI do you have?

I have a 92. It works differently than an 83 if that is what you have.

5. Originally Posted by galactus
Just to do a rough estimate. It will not be as accurate as Excel but it'll do OK.

For the western, use 9.5 as the initial population and the last entry of 23.9.

t=25 since it is 25 years from 1960 to 1985.

$23.9=9.5\cdot 2^{25a}$

Solve for a and we get a=.05324044...

That is pretty close to the Excel method, huh?.

Can you solve the above for a?.

Do the same for the developing world one.
yeah i got the same answer as you did (for the first one by using TI 84 plus - silver edition)

but i just don't know what to do here

$23.9=9.5\cdot 2^{25a}$

i have forgotten the step *sigh*

6. Just divide by 9.5, then use the log properties.

7. ah yeah i got you