Question

The population of a city in 2000 is 50,000. The average increase in population per decade from the previous records of population is 5000 and average percentage increase per decade is 20%. The population of the city based on geometrical increase method, in the year 2020 will be

• A. \( 56,000 \)
• B. \( 64,000 \)
• C. \( 72,000 \)
• D. \( 80,000 \)

Hint:

The geometrical increase method assumes that the population increases at a constant percentage rate per unit of time (e.g., per decade). It is generally suitable for young and rapidly growing cities. In contrast, the arithmetic increase method assumes a constant numerical increase per unit of time and is more suitable for older, established cities.

Answer 1

The Correct Option is C

Step 1: Identify the given data.
Population in year 2000, \( P_0 = 50,000 \) Target year = 2020 Duration from base year to target year = \( 2020 - 2000 = 20 \) years. Number of decades, \( n = \frac{20}{10} = 2 \) decades. Average percentage increase per decade, \( r = 20% = 0.20 \). Note: The information about "average increase in population per decade from the previous records of population is 5000" is relevant for the Arithmetic Increase Method, but since the question specifies "geometrical increase method", we will use the percentage increase.
Step 2: Recall the formula for population projection using the Geometrical Increase Method.
The population at the end of \( n \) decades, \( P_n \), using the geometrical increase method is given by: $$P_n = P_0 (1 + r)^n$$ Where: \( P_0 \) = Initial population \( r \) = Average geometric rate of increase per decade (as a decimal) \( n \) = Number of decades
Step 3: Substitute the values and calculate the population.
Substitute \( P_0 = 50,000 \), \( r = 0.20 \), and \( n = 2 \): $$P_{2020} = 50,000 (1 + 0.20)^2$$ $$P_{2020} = 50,000 (1.20)^2$$ $$P_{2020} = 50,000 \times 1.44$$ $$P_{2020} = 72,000$$
Step 4: Select the correct option.
Based on the calculation using the geometrical increase method, the population in the year 2020 will be \( 72,000 \). $$\boxed{72,000}$$