Question

In 2005, for each of the 14 million people present in a country, $0.028$ was born on $0.008$ died during the year. Using the exponential equation, the number of people present in 2015 is predicted as
A. 25 million
B. 17 million
C. 20 million
D. 18 million

Answer 1

Exponential growth refers to growth in population when limitless resources are available. Exponential growth is used to study the concept of modelling and economical growth. The exponential growth formula reflects the function of exponential growth.
Formula used: $\dfrac{dN}{dt}=rN$. Where, $\dfrac{dN}{dt}=$ Increase in population size in a given time, $r=$ Rate of natural increase in population, and $N=$ Original population size.

Complete answer: The exponential growth formula represents the function of exponential growth. Exponential growth occurs when the population grows continuously with time due to the availability of limitless resources. The exponential growth curve shows a constant growth rate. The exponential growth takes place when the growth rate of the value of a mathematical function depends proportionally on the current value of the function. The result of this involves growth with respect to time, which is an exponential function (function having constant value raised to the power of an argument). Now, in order to find the change in the population density with respect to time $(\dfrac{dN}{dt})$, first, we need to calculate the intrinsic rate of the natural increase, i.e. $r$. In the given question, the $r$ or intrinsic rate of the natural increase is the born rate subtracted from the death rate. That is $r=0.028-0.008=0.02$. The change in time or $dT=2015-2005=10$ years, which implies that we need to find the change in population density in a period of 10 years. $N$ or population density in 2005 was 14 million.
Substituting the above-driven values in the exponential growth formula we get,
$\dfrac{dN}{dt}=0.02\times 14=0.28$ and $dN=0.28\times dt=0.28\times 10=2.8$ million.
Hence, the change in population from 2005 to 2015 will be, $dN=2.8+14=16.8=$ 17 million.
So, option B is the correct answer.

Note: When the growth of a function increases rapidly with time in reference to an increase in total number, it is termed exponential. Exponential growth rates are also used in calculating the bacterial colony population. The colonies grow exponentially until they are provided with a limitless supply of nutrients. Finding the exponential growth rate for colonies gives an estimation of the amount of product that will be obtained.