Understanding Population Growth Rate Calculation

This question asks us to find the annual rate of population increase, represented by R%. We are given the starting population of a town, its population after three years, and we need to determine the percentage rate at which it grew each year.

Population Growth Formula Explained

To solve this, we can use the formula for compound growth, which applies when a quantity increases by a fixed percentage over regular intervals. The formula is:

Final Population = Initial Population $\times$ (1 + $\frac{R}{100}$)$^n$

In this formula:

• The Initial Population is 80,000.
• The Final Population after $n$ years is 1,06,480.
• The number of years, $n$, is 3.
• $R$ is the annual growth rate in percent that we need to calculate.

Step-by-Step Calculation for Rate R

Let's plug the given values into the formula and solve for R:

1. Set up the equation: $1,06,480 = 80,000 \times \left(1 + \frac{R}{100}\right)^3$
2. Isolate the growth factor: To find the factor by which the population increased, divide the final population by the initial population. $\frac{1,06,480}{80,000} = \left(1 + \frac{R}{100}\right)^3$ Simplify the fraction: $\frac{10648}{8000} = \left(1 + \frac{R}{100}\right)^3$ $1.331 = \left(1 + \frac{R}{100}\right)^3$
3. Determine the cube root: The equation shows that the growth factor cubed equals 1.331. To find the growth factor itself, we need to calculate the cube root of 1.331. $(1.331)^{\frac{1}{3}} = 1 + \frac{R}{100}$ Recognizing that $1.1 \times 1.1 \times 1.1 = 1.331$, the cube root of 1.331 is 1.1. $1.1 = 1 + \frac{R}{100}$
4. Solve for R: Now, we rearrange the equation to solve for the rate $R$. Subtract 1 from both sides: $1.1 - 1 = \frac{R}{100}$ $0.1 = \frac{R}{100}$
5. Calculate the percentage rate: Multiply both sides by 100 to find the value of $R$. $R = 0.1 \times 100$ $R = 10$

So, the annual population growth rate R is 10%.