Question

What are exponential growth models?

Answer 1

Hint: We need to explain the concept of exponential growth models here. The relationship between the exponential growth rate and time is exponential. This means that as the time increases linearly, the value of the exponential function grows exponentially. The growth of the function is more and more as time passes by. We explain this concept with the use of an example.

Complete step by step solution:
In order to answer this question, let us first write the mathematical form of an exponential growth model. This is given by the formula $P={{P}_{0}}{{e}^{kt}}.$ Here, P is the value of the function at a particular time t. ${{P}_{0}}$ is the initial value of the function at time t=0. k is called the growth constant which is a constant or factor by which the time is multiplied in order to determine the exponential growth.
This equation says that as the time increases, the growth of the system is large and grows exponentially. We use this in many day-to-day activities for calculations involving growth of population, interest calculations, radioactive decay, etc.
Let us take an example of a bacteria growing in a culture. Its population is assumed to grow exponentially according to the exponential function given by $P\left( t \right)=200{{e}^{0.02t}},$ where 200 is the initial population at a time t=0, where t is the time in minutes. Now we shall calculate the population after 300 minutes.
Substituting the time t as 300 in the given function, we get
$\Rightarrow P\left( 300 \right)=200{{e}^{0.02\times 300}}$
Taking a product of the power of the exponential terms,
$\Rightarrow P\left( 300 \right)=200{{e}^{6}}$
Taking the value of e is 2.718 and substituting in the expression,
$\Rightarrow P\left( 300 \right)=200\times {{2.718}^{6}}$
Taking the product of the terms,
$\Rightarrow P\left( 300 \right)=80685.7586$
This can be approximated as 80,686. Hence, the growth of the bacterial population is exponential and after a time of 300 minutes, its count increases to 80,686.
Hence, we have explained the concept of exponential growth models.

Note:
We need to know the basic concepts of exponents and logarithms in order to understand this question. It is to be noted that the growth of any mathematical function in any manner is used to study and predict the values of the function at a future instant.