The population of a region is growing exponentially. There were 40,000,000 people in 2000 (t = 0)...

Question:

The population of a region is growing exponentially. There were 40,000,000 people in 2000 (t = 0) and 48,000,000 in 2010.

a. Find an expression for the population at any time t, in years.

b. What population would you predict for the year 2020?

c. What is the doubling time?

Exponential growth:

Growth of a quantity means an increase in the number of that quantity over time. The growth can be slow or fast or very very fast. Exponential growth is a type of growth that is slow at the start but becomes very fast thereafter. In exponential growth, the rate of growth at any time is directly proportional to the amount of quantity present at that time.

Exponential growth is mathematically expressed by the following equation;

{eq}\begin{align*} p = p_o e^{rt} \end{align*} {/eq}

where

{eq}\begin{align*} p_o &= \text { initial population, } \\ p &= \text { population at any time t ,and } \\ r &= \text { growth constant. } \end{align*} {/eq}

a.

Given to us is a population with

{eq}\begin{align*} p_o &= 40,000,000 , \\ p &= 48,000,000 , \\ t &= 10 \ years. \end{align*} {/eq}

Substituting these values in the above formula for exponential growth;

{eq}\begin{align*} p &= p_o e^{rt} \\ 48000000 &= 40000000 e^{r \times 10} \\ \frac { 6 } { 5 } &= e^{r \times 10} \\ \end{align*} {/eq}

Taking natural logarithm on both sides of the above equation;

{eq}\begin{align*} ln {\frac { 6 } { 5 } } &= ln { e^{r \times 10} }\\ 0.182 &= 10r \\ r &= 0.0182. \end{align*} {/eq}

Thus, an expression for the population at any time t, in years is as follows;

{eq}\begin{align*} \boxed {p = 40,000,000 e^{0.0182 t}}. \end{align*} {/eq}

b.

Population according to the exponential growth model {eq}p = 40,000,000 e^{0.0182 t} {/eq} for the year 2020 , that is, {eq}t = 20 {/eq} is calculated as follows;

{eq}\begin{align*} p &= 40,000,000 e^{0.0182 t} \\ p &= 40,000,000 e^{0.0182 \times 20 } \\ p &= 57562968.57 \\ p &= 5.75 \times 10^7 \end{align*} {/eq}

Population for the year 2020 is {eq}\begin{align*} \boxed {p = 5.75 \times 10^7}. \end{align*} {/eq}

c.

Doubling time of a population is the time taken by a population to grow to double its initial size.

Given an initial population {eq}p_o {/eq}, the population after doubling time t should be {eq}2 p_o {/eq} . The doubling time t for an exponential growth {eq}r = 0.0182 {/eq} is calculated as follows;

{eq}\begin{align*} p &= p_o e^{rt} \\ 2p_o &= p_o e^{0.0182 t} \\ ln \ 2 &= 0.0182 \times t \\ \frac {0.693}{0.0182} &= t \\ t &= 38.08 \ years. \end{align*} {/eq}

Doubling time of the population is {eq}\begin{align*} \boxed {t = 38.08 \ years}. \end{align*} {/eq}