A geographer models population density D(r) = 1000 × e^(-0.05r) people/km², where r is distance from city center in km. What is the total population

["Understanding Population Density with A Geographer’s Model: Estimating Total Population Using Population Density Function", "Population density plays a crucial role in understanding urban development, resource allocation, and infrastructure planning. One powerful model used by geographers to analyze how population distribution changes across space is the exponential decay density function:", "[\nD(r) = 1000 \ imes e^{-0.05r}\n]\nwhere ( D(r) ) represents population density per square kilometer at a distance ( r ) kilometers from the city center, and ( r ) is measured radially outward from the center.", "### What Does This Model Mean?", "The function shows that population density decreases exponentially as distance from the city center increases. At the city core (( r = 0 )), density peaks at 1,000 people per km², but drops steadily, reflecting the common pattern of urban sprawl and decreasing residential concentration outward.", "This form captures real-world observations: central areas are densely packed, while suburbs and outskirts have fewer people per unit area.", "---", "### How to Calculate Total Population Using Density", "To find the total population within a given area, geographers integrate the density function across the city’s radial extent. Assuming circular symmetry, we integrate over concentric rings of width ( dr ) from ( r = 0 ) to ( r = R ), where ( R ) is the city’s maximum sustainable radius.", "The total population ( P ) is:", "[\nP = \int_0^R D(r) \cdot 2\pi r , dr = \int_0^R \left(1000, e^{-0.05r}\right) \cdot 2\pi r , dr\n]", "Simplify:", "[\nP = 2000\pi \int_0^R r e^{-0.05r} , dr\n]", "---", "### Solving the Integral", "We need to evaluate:", "[\n\int_0^R r e^{-0.05r} , dr\n]", "Using integration by parts with:\n- ( u = r ) → ( du = dr )\n- ( dv = e^{-0.05r} dr ) → ( v = -20 e^{-0.05r} )", "Then:", "[\n\int r e^{-0.05r} dr = -20r e^{-0.05r} + 20 \int e^{-0.05r} dr = -20r e^{-0.05r} - 400 e^{-0.05r} + C\n]", "Evaluate from ( 0 ) to ( R ):", "[\n\int_0^R r e^{-0.05r} dr = \left[ -20r e^{-0.05r} - 400 e^{-0.05r} \right]_0^R\n]", "At ( r = R ):\n[\n-20R e^{-0.05R} - 400 e^{-0.05R}\n]\nAt ( r = 0 ):\n[\n-400\n]", "Thus:", "[\n\int_0^R r e^{-0.05r} dr = \left(-20R e^{-0.05R} - 400 e^{-0.05R}\right) - (-400) = 400 - e^{-0.05R}(20R + 400)\n]", "---", "### Plug Back into Population Formula", "[\nP = 2000\pi \left[ 400 - e^{-0.05R}(20R + 400) \right]\n]", "For an infinite-radius city, ( R \ o \infty ), and ( e^{-0.05R} \ o 0 ), simplifying to:", "[\nP = 2000\pi \ imes 400 = 800,!000\pi\n]", "[\nP \approx 800,!000 \ imes 3.1416 = 2,!513,!274 \ ext{ people (approximately)}\n]", "---", "### When Is the Model Valid?", "This infinite integral assumes unbounded urban sprawl, which is unrealistic. In practice, cities have physical or administrative boundaries. To estimate real populations, geographers apply the model to a defined radius ( R ), say 20–50 km, adjusting ( R ) based on urban limits.", "For ( R = 30 ) km:", "[\nP = 2000\pi \left[ 400 - e^{-1.5} (600 + 400) \right] \approx 2000\pi \left[ 400 - 0.2231 \ imes 1000 \right]\n]\n[\nP \approx 2000\pi \ imes (400 - 223.1) = 2000\pi \ imes 176.9 \approx 1,!111,!760\n]", "---", "### Summary", "- The model ( D(r) = 1000 \cdot e^{-0.05r} ) projects how population density declines from urban centers outward.\n- Total population depends on effective city radius ( R ), computed via integral:\n[\nP = 2000\pi \left(400 - e^{-0.05R}(20R + 400)\right)\n]\n- For infinite cities: ( P \approx 2.513 \ imes 10^6 ) people.\n- For realistic ( R ): substitute radius to estimate population.", "Esteeming population through mathematical modeling helps city planners balance housing, transport, and services with spatial reality.", "---", "Key Takeaways:\n- Exponential models capture realistic urban population patterns.\n- Integral calculus connects local density to global population.\n- Adjust model radius to reflect real-world city limits.", "Optimize density data with geospatial modeling to support sustainable urban growth."]