# Population growth help

A population grows at a constant relative rate. After 10 months the population grows to 72, and after 18 months the population grows to 96. Find C and k.

the answer I got was c=50.25 and k=.0359

can somebody just double check for me? I dont know if the method I was used was correct.

## Answers and Replies

CompuChip
Science Advisor
Homework Helper
If you have doubts about your method, you better post the method :)

I get (up to two decimals) the same answers, but that might be a co-incidence.

[Also, please be clear about your notation. I just assumed that C would be the initial population and k the rate of growth.]

ok heres the method I used, its very long and we haven't had to solve these type of problems this way in class so I was apprehensive about it

C = Initial Population K = Relative Growth Rate

P(10) = Ce^(10k) = 72 P(18) = Ce^(18k) = 96
e^(10k) = 72/C e^(18k) = 96/C
10k = ln(72/C) 18k = ln(96/C)
k = (ln(72/C))/10 k = (ln(96/C))/18

k = k
(ln(72/C))/10 = (ln(96/C))/18

18(ln(72/C)) = 10(ln(96/C))
18ln72 - 18lnC = 10ln96 - 10lnC
76.98 - 18lnC = 45.64 - 10lnC
-10lnC + 18lnC = 76.98 - 45.64
8lnC = 31.34
lnC = 3.92
C = 50.4 (I got 50.25 because I didnt round so I could get a more accurate answer)

then I plugged in C to find k

(ln(72/50.4))/10 = .0356 = k

was my method correct? I felt like I was making a simple problem, harder than what it was, but idk

shouldnt C be a whole number since it is a population?

CompuChip
Science Advisor
Homework Helper
By the looks of it, your method is correct, although it is not the most clear way to write down what you are doing. In particular, I was wondering where the exponential is coming from on the first line, although I think in the end it doesn't matter because taking the wrong base number (e instead of something else) produces a common factor which drops out when you equate the two.

Here is what I would expect.
The formula for growth at a constant relative rate is
P(n) = C (1 + k)^n
because for n = 0 the population is C, and for each time step you have to multiply the previous population P(n - 1) by (1 + k).

Then it is given that
P(10) = 72 = C (1 + k)^(10)
P(18) = 96 = C (1 + k)^(18)
which gives you two equations in two unknowns.

What you could do is divide them, and get
$$\frac{P(18)}{P(10)} = \frac{96}{72} = \frac{C (1 + k)^{18}}{C (1 + k)^{10}} = (1 + k)^8$$
and you see that C drops out. So now it's easy to solve for k:
$$(1 + k)^8 = 4/3$$
so
$$1 + k = \sqrt{4/3} = 1,0366...$$
$$\qquad\implies k = 0,0366...$$

To find C, you only need to plug it back in to one of the equations, for example
P(10) = 72 = C * (1,0366...)^(10)
(try not to round, e.g. if you use a calculator try to use it's ANS function to plug in 1 + k) and so
C = 72 / (1,0366...)^(10) = 50.25....

Note how the formula one starts with is more intuitive (I see an exponential in your formula, while it says the growth is at constant relative rate; my formula explicitly has this behaviour by multiplying by the growth percentage at every step) and easier to work with (you don't need logarithms, just some root).

That C and P(n) are in general not integer numbers is common in such exercises. You need to remember that this is merely a model for some "real" process, which makes assumptions (constant growth), tries to describe something discrete (counting the population every month) by a continuous function (you can calculate C (1 + k)^n for any n), and all we want of it is to give us more or less the right numbers at n = 0, 1, 2, ... although we will never be able to use it to predict any exact historic or future data.