Mathematical Science

Population Growth Calculation

Problem Statement The population of a city is increasing by 10% annually. Given the current population is 20,000, we need to find the population of the city after 2 years.

Mathematical Representation

To determine the future population, we will use the formula for compound interest, which is applicable here since the growth rate is compounded annually. The formula for population growth is:

$P_{}=P_{}\times (1+r{)}^{n}P\_\{\backslash text\{future\}\}\; =\; P\_\{\backslash text\{present\}\}\; \backslash times\; (1\; +\; r)^n$Pfuture​=Ppresent​×(1+r)n

where:

• $P_{}P\_\{\backslash text\{future\}\}$Pfuture​ = population after $nn$n years
• $P_{}P\_\{\backslash text\{present\}\}$Ppresent​ = present population
• $rr$r = annual growth rate (expressed as a decimal)
• $nn$n = number of years

In this case:

• $P_{}=20,000P\_\{\backslash text\{present\}\}\; =\; 20,000$Ppresent​=20,000
• $r=10\mathrm{\%}=0.10r\; =\; 10\backslash \%\; =\; 0.10$r=10%=0.10
• $n=2n\; =\; 2$n=2

Applying the Formula

1. Substitute the values into the formula:

   $P_{}=20,000\times (1+0.10{)}^{2}P\_\{\backslash text\{future\}\}\; =\; 20,000\; \backslash times\; (1\; +\; 0.10)^2$Pfuture​=20,000×(1+0.10)2

2. Calculate $(1+0.10{)}^2(1\; +\; 0.10)^2$(1+0.10)2:

   $(1+0.10{)}^2=1.10^2(1\; +\; 0.10)^2\; =\; 1.10^2$(1+0.10)2=1.102

   $1.10^2=1.211.10^2\; =\; 1.21$1.102=1.21

3. Multiply by the present population:

   $P_{}=20,000\times 1.21P\_\{\backslash text\{future\}\}\; =\; 20,000\; \backslash times\; 1.21$Pfuture​=20,000×1.21

   $P_{}=24,200P\_\{\backslash text\{future\}\}\; =\; 24,200$Pfuture​=24,200

Conclusion

The population of the city after 2 years will be 24,200.

Recent Examples and Context

• Urban Planning and Development: In recent years, city planners and administrators often use similar calculations to forecast future population growth for infrastructure development and resource allocation. For example, the city of Bangalore, India, has used growth models to plan for future urban expansion and public services.
• Corporate Business Models: Companies, especially those in retail and real estate, also use compound growth calculations to project future market sizes and demand. For instance, retail chains might use such projections to plan new store openings based on anticipated population increases.

Verification

To ensure the accuracy of the calculation:

• Annual Growth Check: Verify by calculating the population at the end of each year. For the first year, the population would be $20,000\times 1.10=22,00020,000\; \backslash times\; 1.10\; =\; 22,000$20,000×1.10=22,000. For the second year, applying the same growth rate to 22,000 gives $22,000\times 1.10=24,20022,000\; \backslash times\; 1.10\; =\; 24,200$22,000×1.10=24,200, confirming the final result.

Thus, the projected population of the city after 2 years, given the 10% annual increase, is indeed 24,200.