Question

The population of a city in the year 2001, 2011, 2021 were recorded as 52,000, 76,000, and 1,20,000 respectively. Calculating the average growth rate using geometric mean, the estimated population of the city for 2031 using the geometric increase method is _________. (rounded off to the nearest integer)

Hint:

When using geometric mean to calculate growth rates, ensure that you apply the formula correctly and round off the results as needed.

Answer 1

Given:
Population in 2001 (\(P_1\)) = 52,000
Population in 2011 (\(P_2\)) = 76,000
Population in 2021 (\(P_3\)) = 1,20,000
Step 1: The formula for the average growth rate using the geometric mean is: \[ r = \left( \frac{P_2}{P_1} \times \frac{P_3}{P_2} \right)^{\frac{1}{2}} - 1 \] Substitute the given values: \[ r = \left( \frac{76,000}{52,000} \times \frac{1,20,000}{76,000} \right)^{\frac{1}{2}} - 1 \] \[ r = \left( 1.4615 \times 1.5789 \right)^{\frac{1}{2}} - 1 \] \[ r = \left( 2.3086 \right)^{\frac{1}{2}} - 1 = 1.519 - 1 = 0.519 \] Step 2: The estimated population for 2031 can be found using the formula for geometric increase: \[ P_{{2031}} = P_3 \times (1 + r)^{10} \] Substitute the values: \[ P_{{2031}} = 1,20,000 \times (1 + 0.519)^{10} \] \[ P_{{2031}} = 1,20,000 \times (1.519)^{10} = 1,20,000 \times 1.957 = 179,000 \] Thus, the estimated population of the city for 2031 is 179,000.