Population Biology Question

Oh joy. We just came to my most favorite chapter in AP Environmental Science! (sarcasm) Anyway I'm good with every question I've been presented with so far except this one:

3) Given a growth rate of 3 percent per year, how long will it take for a population of 100,000 individuals to double? How long will it take to double when the population reaches 10 million?

My teacher said in order to solve it, you have to use one of the two mathematical growth formulas, either exponential or logistic.

Exponential: dN/dt = rN, that is,the change in the number of individuals (dN) per change in time (dt) equals the rate of growth ® times the number of individuals in the population (N).

Logistic: dN/dt = rN(1- N/K), where the change in numbers over time (dN/dt) equals the exponential growth rate (rN) times the portion of the carrying capacity (K) not already taken by the current population size (N).

This is the thing though, there is no given carrying capacity in the question so I assumed that using the logistic equation was not an option, until I remembered that exponential growth has no limits. >.< Advice on how to solve this question would be appreciated.

Do you know recurrence relations, or geometric series?

Assuming you know recurrence:



That's what we'll work with.

As you might see, this is a linear recurrence relation. We can solve it:



And from there on, use logarithms to find what you want. (In this case, you want these two:
)

Noah

Do you know recurrence relations, or geometric series?

Yea.....no......

I'm only in 10th grade in Algebra 2.

I covered that in the second half of tenth grade, in trig 2.
You should know it by the end of this year, but they're pretty simple to understand.
http://en.wikipedia.org/wiki/Recurrence_relation
http://en.wikipedia.org/wiki/Geometric_progression

The geometric series on wikipedia is about infinite sums, don't get confused.

Yea.....no......

I'm only in 10th grade in Algebra 2.

Okay, then we'll just have to teach you geometric series!

You see that we start at "year 0", and at year 1, you have 100 000 dollars plus those three percents, which is in total 103 000 dollars. This can be written as 100 000 * 1.03, right?

Now, what about the next year? Well, that is 103 000 dollars plus those three percents, which is in total 106 090 dollars. That can be written as 103 000 * 1.03, right? But hey, 103 000 can be written as 100 000 * 1.03! Therefore, we can say that, on the second year, you have 100 000 * 1.03 * 1.03 dollars in the bank account. Or, to make it simpler, 100 000 * 1.03^2 dollars.

Now, this is true for all (positive) years! Therefore, you can say that "at year n, I'll have 100 000*1.03^n dollars in my bank account"

To find the answers (if you don't know log), just multiply by 1.03 until you've bypassed what you wanted. Just be sure to count all the times you've multiplied by 1.03, as I tend to forget that.

Noah

Okay, then we'll just have to teach you geometric series!

You see that we start at "year 0", and at year 1, you have 100 000 dollars plus those three percents, which is in total 103 000 dollars. This can be written as 100 000 * 1.03, right?

Now, what about the next year? Well, that is 103 000 dollars plus those three percents, which is in total 106 090 dollars. That can be written as 103 000 * 1.03, right? But hey, 103 000 can be written as 100 000 * 1.03! Therefore, we can say that, on the second year, you have 100 000 * 1.03 * 1.03 dollars in the bank account. Or, to make it simpler, 100 000 * 1.03^2 dollars.

Now, this is true for all (positive) years! Therefore, you can say that "at year n, I'll have 100 000*1.03^n dollars in my bank account"

To find the answers (if you don't know log), just multiply by 1.03 until you've bypassed what you wanted. Just be sure to count all the times you've multiplied by 1.03, as I tend to forget that.

Noah

This seems to make sense to me (I'll solve it this way), but how would I set up the problem using the exponential formula? Because I really don't think my teacher will want us to use anything more than basic Algebra 1 on problems like this.

Thats something I haven't heard in a while "my teacher wanting it x way", hell, if you find out a better way to do an exercise, you should do it that way, if your teacher asks how you did it, you either explain it or tell him you found a source that helped you.

Who knows, he might congratulate you, or be an ass about it.

Oh God....Wow .....I totally forgot about the Rule of 70. Nevermind guys, this problem looks a WHOLE lot easier now. I know how to solve it. Thanks for other methods of solving this problem though....