In , koalas were introduced on Kangaroo Island off the coast of Australia. In ,

Question

In , koalas were introduced on Kangaroo Island off the coast of Australia. In , the population was . By , the population had grown to , prompting a debate on how to control their growth and avoid koalas dying of starvation1. Assuming exponential growth, find the (continuous) rate of growth of the koala population between and . Round your answer to one decimal place. The continuous growth rate is ____% Find a formula for the population as a function of the number of years since . Assume that denotes the number of years. The formula is Estimate the population in the year . Round your answer to the nearest thousand koalas. The population in the year will be approximately ______ thousand koalas

Explanation / Answer

p = P_0*(1 + x)^t

when t = 0, p = 5000
5000 = P_0*1

when t = 9, p = 27000
27000 = 5000*(1 + x)^9
(1 + x)^9 = 27000/5000
1 + x = (27/5)^(1/9)
x = (27/5)^(1/9) - 1
x = 0.206
x = 20.6%

Define variables.

Let t denote the number of years after 1996.
Let P(t) denote the population (in thousands) after t years.
Let k denote the continuous growth rate.

Then a model for P(t) is

P(t) = P(0) e^(kt)

You are given P(0) = 5 and P(2005-1996) = P(9) = 27. Putting these in the model,

27 = 5e^(9k)

Solve this for k.

27/5 = e^(9k)
ln(27/5) = 9k
ln(27/5)/9 = k

To one decimal place, k = 0.2 (to more decimal places, it's 0.187377661)

You could get a general formula by following precisely the same steps as for the particular one:

P(t) = P(0) e^(kt)
P(t)/P(0) = e^(kt)
ln(P(t)/P(0)) = kt
ln(P(t)/P(0)) / t = k

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