Question

The population of a mosquito population obeys the law of uninhibited growth.If there are 500 mosquito initially and there are 800 after 1 day. How long is it until there are 7000 mosquito?Round your answer to the nearest tenth.

Answer 1

The law of uninhibited growth is express as:

`N(t)=N_0e^(kt)`

where N(t) is the population at time t, N0 is the initial population, k is the growth rate and t is the time. In this case we know that after one day, t=1, the population is 800 and that the initial population was 500; plugging these values and solving for k we have:

`\begin{gathered} 800=500e^k \\ e^k=(800)/(500) \\ \ln e^k=\ln(8)/(5) \\ k=\ln(8)/(5) \end{gathered}`

Now that we have the growth rate, we know that the population growth in this case can be express as:

`N(t)=500e^{(\ln(8)/(5))t}`

We want to know the time it takes for the population to be 7000, to find it we equate our expression to this value and solve for t:

`\begin{gathered} 7000=500e^{(\ln(8)/(5))t} \\ e^{(\ln(8)/(5))t}=(7000)/(500) \\ \ln e^{(\ln(8)/(5))t}=\ln14 \\ (\ln(8)/(5))t=\ln14 \\ t=(\ln14)/(\ln(8)/(5)) \\ t=5.6 \end{gathered}`

Therefore, it takes 5.6 days for the population to reach 7000 individuals.