- #1

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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.

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- Thread starter r_swayze
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- #1

- 66

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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.

- #2

CompuChip

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Homework Helper

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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.]

- #3

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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?

- #4

CompuChip

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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

[tex]\frac{P(18)}{P(10)} = \frac{96}{72} = \frac{C (1 + k)^{18}}{C (1 + k)^{10}} = (1 + k)^8[/tex]

and you see that C drops out. So now it's easy to solve for k:

[tex](1 + k)^8 = 4/3[/tex]

so

[tex]1 + k = \sqrt[8]{4/3} = 1,0366...[/tex]

[tex]\qquad\implies k = 0,0366...[/tex]

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.

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