Question

The population of a certain species of insects increases exponentially according to the model, P(0) = 952e0.12t where P is the number of insects and t is the time in weeks since the original colony settled in the farm nearby. A. How many insects are in the original colony? B. How many insects would you expect to be in the colony after 20 weeks if the population growth follows the model? Round off to the nearest whole number. C. How long will it take for the population to reach 20000?

Answer 1

Answer:

A) 952 insects are in the original colony

B) 10494 insects are in the colony after 20 weeks

C) 25 weeks

Explanations:

The model representing the population of the species of insects

$P(t)=952e^{0.12t}$

An exponential growth is of the form:

$P(t)=P_0e^{kt}$

where P₀ is the original population

t is the time taken in weeks

Comparing the two equations:

$P_0=\text{ 952}$

952 insects are in the original colony

B. The number of insects that will be in the colony after 20 weeks

Substituting t = 20 into the function given

$\begin{gathered} P=952e^{0.12(20)} \\ \text{P = }10494 \end{gathered}$

C) If the population, P = 20000

$\begin{gathered} 20000=952e^{0.12t} \\ e^{0.12t}=\text{ }\frac{20000}{952} \\ e^{0.12t}=21.008 \\ 0.12t\text{ = ln(21.008)} \\ 0.12t\text{ = }3.04 \\ t\text{ = }\frac{3.04}{0.12} \\ t\text{ = }25 \end{gathered}$