Math Challenges

Submissions for Problem #16

Problem #16

New Zealand's rabbit population was estimated to be 25 million in 2010 and 35 million in 2020. Model the population of the rabbits with R(t) where t = 0 corresponds to the year 2010.

What is an appropriate value for the growth-rate k?
What would you predict the population to be in the year 2030?

\[ \frac{dR}{\differentialD t}=kR \]

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zapwai

Solution: \( \displaylines{R\left(0\right)=25,R\left(10\right)=35\\ \text{Since }R\left(t\right)=Ce^{kt}\text{ we have }R\left(0\right)=C=25,R\left(10\right)=25e^{k\left(10\right)}=35=>\\ \frac{35}{25}=e^{10k}\Rightarrow\ln\left(\frac{35}{25}\right)=10k\Rightarrow k=\frac{1}{10}\ln\left(\frac{35}{25}\right)\approx0.034\\ R\left(t\right)=25e^{0.034t},\text{ and evaluating when }t=20\text{ we can predict population in 2030 as}\\ R\left(20\right)=25e^{0.034(20)}=49} \)

Explanation:
k = 0.034 is the growth-rate and we expect roughly 49 million in 2030!