The formula for exponential growth is expressed as
y = a(1 + r)^t
where
y is the population after t years
ais the initial population
r is the growth rate
t is the number of years
From the information given,
a = 350
r = 14% = 14/100 = 0.14
By substituting these values into the equation, the function to model th growth rate is
y = 350(1 + 0.14)^t
y = 350(1.14)^t
We would substitute values of t into the equation to determine corresponding values of y. We have
For t = 2, y = 350(1.14)^2 = 454.86
For t = 6, y = 350(1.14)^6 = 768.24
For t = 10, y = 350(1.14)^10 = 1297.52
We would plot these values of y on the y axis and the corresponding values of t on the x axis. The graph is shown below
The time when the population will reach 20000 would be the value of x at the point where y = 20000 on the graph. It is around y = 30.87
Thus, the population will reach 20000 in approximately 31 years
We can confirm this by solving the equation. We have
20000 = 350(1.14)^t
20000/350 = 1.14^t
57.14 = 1.14^t
Taking natural log of both sides, we have
ln 57.14 = ln1.14^t = tln1.14
t = ln 57.14/ln1.14
t = 30.87505