Population Growth Rate Calculation

This problem involves calculating the annual rate of population increase (R%) over a period of three years, given the initial and final populations. We can solve this using the compound growth formula.

Understanding the Compound Growth Formula

The formula for compound growth is:

$ P_n = P_0 \left(1 + \frac{R}{100}\right)^n $

Where:

• $P_n$ is the population after $n$ years.
• $P_0$ is the initial population.
• $R$ is the annual growth rate in percent.
• $n$ is the number of years.

Applying the Formula to the Problem

From the question, we have:

• Initial population, $P_0 = 80,000$
• Population after 3 years, $P_3 = 1,06,480$
• Number of years, $n = 3$
• Annual growth rate = $R\%$

Step-by-Step Calculation

1. Substitute the known values into the formula:

   $ 1,06,480 = 80,000 \left(1 + \frac{R}{100}\right)^3 $

2. Isolate the growth factor term:

   Divide both sides by the initial population (80,000):

   $ \frac{1,06,480}{80,000} = \left(1 + \frac{R}{100}\right)^3 $

   $ 1.331 = \left(1 + \frac{R}{100}\right)^3 $

3. Find the cube root of both sides:

   To find the value of $\left(1 + \frac{R}{100}\right)$, we take the cube root of 1.331:

   $ \sqrt[3]{1.331} = 1 + \frac{R}{100} $

   Since $1.1 \times 1.1 \times 1.1 = 1.331$, the cube root of 1.331 is 1.1.

   $ 1.1 = 1 + \frac{R}{100} $

4. Solve for R:

   Subtract 1 from both sides:

   $ 1.1 - 1 = \frac{R}{100} $

   $ 0.1 = \frac{R}{100} $

   Multiply both sides by 100:

   $ R = 0.1 \times 100 $

   $ R = 10 $

Conclusion

The value of R, representing the annual population growth rate, is 10%.