Exponential growth and decay are rates. They represent the change in some quantity through time.
Exponential Growth
Exponential growth is any increase in a quantity P.
P(t)= Poe^(kt)
Where Po is the initial quantity, t is time, k is a constant, P(t) is the quantity after time t, and e^x is the exponential function.
Exponential Decay
Exponential Decay is an decrease in a quantity P.
P(t)= Poe^(-kt)
Where Po is the initial quantity, t is time, k is a constant, P(t) is the quantity after time t, and e^x is the exponential function.
Exponential Growth and Decay Models
Po<k is a graphical interpretation of exponential decay.
Po>k is a graphical interpretation of exponential growth.
Po=k is a graphical interpretation of constant exponential growth/decay.
1. If a city has a population of 340 people, and if the population grows continuously at an annual rate of 2.3%, what will the population be in 6 years?
We are given Po= 340 people and k= .023.
P(t)= 340e^(.023t)
When t= 6 years
P(t)= 340e^(.023*6)= ~390 people
2. If China has a population of 420 tigers, and if the population doubles every 9 years, what will the population be in 7 years?
We are given Po= 420 and t= 7 and P(9)= 840.
A doubling time of 9 years means that 840= P(9)= Poe^(kt), and follows that 2= e^(9k).
Take the natural log of both sides to get ln2=9k.
So k=(ln2)/9
Hence, P(7)= 420e^(7k)= 420e^((7ln2)/9)= ~720 tigers
