Question

Two years ago the population of a town was 40000. The population of the town at present has reached 44100. Calculate the population growth rate of the city.​

Answer 1

Answer:

5% annual population growth rate

Step-by-step explanation:

Let the percent the population grows by be $X\%$. The total population, $f(x)$, after $t$ years can be modeled by the function:

$f(x)=40,000\cdot (\frac{X}{100}+1)^t$

Why?

Let's take a look at a simple example. If we said a number $n$ grew by 10%, we could represent the number after it grew by multiplying $n$ by $1.10$. This is because growing by 10% is equivalent to taking $100\%+10\%=110\%$ of that number and we convert a percentage to a decimal by dividing by 100.

Therefore, if the population grew $X\%$, we would divide it by 100 to convert it to a decimal, then add 1 (100%) and raise to the power of $t$ (number of years) to multiply by the initial population of 40,000 to get the total population after $t$ years.

Since the population of the town after two years is 44,100, substitute $f(x)=44,100$ and $t=2$ into $f(x)=40,000\cdot (\frac{X}{100}+1)^t$:

$44,100=40,000\cdot (\frac{X}{100}+1)^2,\\\\(\frac{X}{100}+1)^2=\frac{44,100}{40,000},\\\\(\frac{X}{100}+1)^2=1.1025,\\\\(\frac{X}{100}+1)^2=\pm \sqrt{1.1025},\\\\\begin{cases}\frac{X}{100}+1=1.05,\frac{X}{100}=0.05, X=\boxed{5\%},\\*\text{negative case is extraneous since X must be positive}\end{cases}$

Therefore, the city has an annual population growth rate of 5%.