Question

Find a formula for P = f(t), the size of the population that begins in year t = 0 with 2070 members and decreases at a 3.9% annual rate. Assume that time is measured in years. P = f(t) =

Answer 1

$P=f(t)=P(1-0.039)^t$

Explanation

Step 1

let

$P=f(t)$

where P represent the population, and t represents the time in years

so,

when t=0, P=2070

$\begin{gathered} P=f(t) \\ 2070=f(0) \end{gathered}$

Step 2

if the population decrease 3.9% every year,in decimal form

$\begin{gathered} \text{3}.9\text{ =3.9/100= 0.039} \\ \end{gathered}$

so,after 1 year the population is

$\begin{gathered} P_1=P(1-0.039)\rightarrow Eq1 \\ P_1=P(0.961) \\ P_1=2070(0.961) \\ P_1=1989.27 \end{gathered}$

now, after the 2 years

$\begin{gathered} P_2=P_1(1-0.039)=P(1-0.039)(1-0.039)=P(1-0.039)^2 \\ \end{gathered}$

now, after 3 years

$P_3=P_2(1-0.039)=P(1-0.039)(1-0.039)(1-0.39==P(1-0.039)^3$

now, we can see the function

$\begin{gathered} P(1-0.039)^t\rightarrow P_f=P(1-0.039)^t \\ f(t)=P(1-0.039)^t \end{gathered}$

I hope this helps you