A geographer is mapping a region using GIS and finds that the population density increases exponentially from 50 people/km² at 10 km from a city center to 800 people/km² at 20 km. Assuming exponential growth, what is the density at 15 km?

["Title: Exponential Population Density Growth Mapped Using GIS: Calculating Density at 15 km from City Center", "Meta Description: Using GIS to model population density, a geographer finds exponential growth from 50 people/km² at 10 km to 800 people/km² at 20 km. Discover how exponential functions reveal urban expansion patterns.", "---", "### Understanding Exponential Growth in Urban Population Density", "When mapping human settlements, geographers often rely on Geographic Information Systems (GIS) to model how population density changes across space. Recent fieldwork using GIS technology reveals a striking pattern: population density in a region near a central city follows an exponential trend—not a steady increase—meaning density grows rapidly with distance from the urban core.", "A geographer’s study shows that starting from 50 people per square kilometer just 10 km from the city center, the density rises exponentially to 800 people/km² at 20 km. This kind of spatial distribution is classic for urban expansion, where development gradually tapers off as distance from the core increases.", "---", "### Modeling the Exponential Relationship", "Population density as a function of distance from the city center can be expressed mathematically using exponential growth:", "[\nD(x) = D_0 \cdot e^{kx}\n]", "Where:\n- ( D(x) ) = population density at distance ( x ) km from the city center\n- ( D_0 ) = density at the reference point (10 km = 50 people/km²)\n- ( k ) = growth rate constant\n- ( x ) = distance in kilometers from the city center", "We are given:\n- At ( x = 10 ) km, ( D(10) = 50 )\n- At ( x = 20 ) km, ( D(20) = 800 )", "Using the first data point:\n[\nD(10) = D_0 \cdot e^{10k} = 50\n]\nSo,\n[\nD_0 = 50 \cdot e^{-10k}\n]", "Substitute into the second equation:\n[\n800 = D_0 \cdot e^{20k} = (50 \cdot e^{-10k}) \cdot e^{20k} = 50 \cdot e^{10k}\n]", "Divide both sides by 50:\n[\n16 = e^{10k}\n]", "Take the natural logarithm:\n[\n\ln(16) = 10k \Rightarrow k = \frac{\ln(16)}{10} = \frac{4\ln(2)}{10} = 0.4\ln(2) \approx 0.2773\n]", "Now find ( D_0 ):\n[\nD_0 = 50 \cdot e^{-10k} = 50 \cdot e^{-\ln(16)} = 50 \cdot \frac{1}{16} = 3.125\n]", "Thus, the full model is:\n[\nD(x) = 3.125 \cdot e^{0.2773x}\n]", "---", "### Estimating Density at 15 km", "Now compute ( D(15) ):\n[\nD(15) = 3.125 \cdot e^{0.2773 \ imes 15} = 3.125 \cdot e^{(4.16475)}\n]", "Using ( e^{4.16475} \approx 64 ) (since ( \ln(64) = 4.1597 ), close enough),\n[\nD(15) \approx 3.125 \ imes 64 = 200\n]", "So, the population density at 15 km from the city center is approximately 200 people/km².", "---", "### Real-World Implications", "This exponential model shows that urban populations aren’t evenly spaced—instead, they cluster intensely near the core and thin out rapidly. GIS-powered mapping with exponential functions helps planners anticipate infrastructure needs, optimize transportation, and manage environmental impacts.", "Understanding these spatial patterns supports sustainable development, particularly in fast-growing metropolitan regions where land use and resource allocation are critical.", "---", "### Conclusion", "Using exponential modeling in GIS, geographers uncover the true shape of urban expansion. From 50 people/km² at 10 km to 800 at 20 km, the density rises from 50 to 800 in just 10 km—proving that real-world human settlements often expand in accelerating waves. The density at 15 km, calculated as 200 people/km², reflects this smooth yet dynamic transition.", "By applying mathematical precision to geography, we gain deeper insight into how cities grow—and how best to plan for their future.", "---", "Keywords: GIS population density, exponential growth model, urban expansion, geographer fieldwork, population mapping, exponential density calculation, 15 km from city center, spatial analysis, urban geography", "For further reading: GIS tools in urban planning, population density modeling, exponential spatial growth analysis."]