Question

The population of a certain species of insects in a nearby farm increases exponentially according to themodel, () = 1840^0.145where P is the number of insects and t is the time in weeks since the originalcolony settled in the farm.A. Find the number of insects in the original colony that settled in the farm.B. Find the rate of increase of the insect population for each week that passed by.C. How many insects would you expect to be in the colony after 16 weeks if the population growthfollows the model? Round off to the nearest whole number.D How long will it take for the population to reach 12000

Answer 1

Continuous Exponential Model

It's an equation that models the growth of a certain variable in time. If P(t) is the number of insects in a colony at a time t, and Po is the initial number of insects of the colony, then the model can be written as:

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

Where k is the growth rate.

We are given the model equation:

$P(t)=1840e^{0.145t}$

Where the time is in weeks.

A. This corresponds to an original number of insects of Po = 1840

B. The rate of increase can also be determined with the equation. k = 0.145. This corresponds to a rate of 14.5% each week.

C. Now we find the number of insects for t = 16:

$\begin{gathered} P(16)=1840e^{0.145*16} \\ Calculating: \\ P(16)=1840e^{2.32} \\ P(16)=18723 \end{gathered}$

It's expected to be 18723 insects in the colony after 16 weeks.

D. For the colony to have 12000 insects, we need to solve the equation:

$1840e^{0.145t}=12000$$1840e^{0.145t}=12000$

Dividing by 1840:

$e^{0.145t}=\frac{12000}{1840}$

Applying logs on each side:

$0.145t=\log\frac{12000}{1840}$

Dividing by 0.145:

$t=\frac{\log\frac{12000}{1840}}{0.145}$

Calculating:

t ≈ 13 weeks

It will take approximately 13 weeks for the population to reach 12000