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:
\n Population in 2001 (\(P_1\)) = 52,000
\n Population in 2011 (\(P_2\)) = 76,000
\n Population in 2021 (\(P_3\)) = 1,20,000
\n\nStep 1: Calculate the average annual growth rate using the geometric mean formula:\n\[\nr = \left( \frac{P_2}{P_1} \times \frac{P_3}{P_2} \right)^{\frac{1}{2}} - 1\n\]\nSubstitute the given population figures:\n\[\nr = \left( \frac{76,000}{52,000} \times \frac{1,20,000}{76,000} \right)^{\frac{1}{2}} - 1\n\]\nPerform the calculations:\n\[\nr = \left( 1.4615 \times 1.5789 \right)^{\frac{1}{2}} - 1\n\]\n\[\nr = \left( 2.3086 \right)^{\frac{1}{2}} - 1 = 1.519 - 1 = 0.519\n\]\n\nStep 2: Estimate the population for 2031 using the geometric increase formula:\n\[\nP_{{2031}} = P_3 \times (1 + r)^{10}\n\]\nInsert the known values:\n\[\nP_{{2031}} = 1,20,000 \times (1 + 0.519)^{10}\n\]\nCalculate the estimated population:\n\[\nP_{{2031}} = 1,20,000 \times (1.519)^{10} = 1,20,000 \times 1.957 = 179,000\n\]\n\nThe estimated population for the city in 2031 is 179,000.