The population of a city grows exponentially according to \( P(t) = 50000 \cdot e^{0.03t} \), where \( t \) is in years. How long will it take for the population to double?

["How Long Will It Take for the City’s Population to Double? Using Exponential Growth (( P(t) = 50000 \cdot e^{0.03t} ))", "Urban populations are expanding rapidly around the world, and understanding how quickly a city’s population grows is essential for planning infrastructure, healthcare, and services. One powerful model used by demographers is exponential growth, represented mathematically as:", "[\nP(t) = P_0 \cdot e^{rt}\n]", "where:\n- ( P(t) ) is the population at time ( t ),\n- ( P_0 ) is the initial population,\n- ( r ) is the annual growth rate (as a decimal),\n- ( t ) is time in years.", "In this article, we explore a real-world example: a city whose population follows the model:", "[\nP(t) = 50000 \cdot e^{0.03t}\n]", "Here, the initial population ( P_0 = 50,000 ), and the annual growth rate ( r = 0.03 ), or 3%. A key question many urban planners and residents ask is: How long will it take for the population to double?", "### The Doubling Time Formula", "To find the doubling time—time ( T ) when the population becomes double ( P_0 ), we set:", "[\nP(T) = 2 \cdot P_0\n]", "Substituting into the formula:", "[\n2P_0 = P_0 \cdot e^{0.03T}\n]", "Divide both sides by ( P_0 ):", "[\n2 = e^{0.03T}\n]", "Now take the natural logarithm of both sides:", "[\n\ln(2) = \ln(e^{0.03T}) = 0.03T\n]", "Solve for ( T ):", "[\nT = \frac{\ln(2)}{0.03}\n]", "Using ( \ln(2) \approx 0.6931 ):", "[\nT = \frac{0.6931}{0.03} \approx 23.10 \ ext{ years}\n]", "### Conclusion", "The city’s population will double approximately every 23.1 years, based on the current exponential growth rate of 3% per year.", "This insight helps city officials anticipate future challenges—such as housing demand, transportation needs, and resource allocation—allowing for proactive urban planning. While exponential growth does not continue indefinitely in reality (due to factors like resource limits and policy changes), understanding the doubling time provides a valuable benchmark for sustainable development.", "Keywords: exponential population growth, doubling time formula, urban planning, e^(0.03t) population model, 3% annual growth, demography, city expansion, population doubling time.", "---", "Unlock your city’s future now—understanding exponential growth empowers smarter decisions."]