Question: A hydrologist observes a circular drainage basin with area $ 100\pi \, \text{km}^2 $. If the radius decreases by 2 km due to sedimentation, by how many square kilometers does the area decrease?

["Understanding Area Loss in a Circular Drainage Basin: A Hydrologist’s Perspective", "When studying rainfall and watershed dynamics, hydrologists rely on precise measurements of drainage basins to model water flow and predict flooding. A critical aspect involves calculating how changes in basin geometry affect surface water collection and runoff. One key scenario involves a circular drainage basin experiencing shrinkage due to sedimentation, which directly reduces its capacity. In this case, a hydrologist observes a circular drainage basin with an initial area of $ 100\pi , \ ext{km}^2 $; if sedimentation gradually reduces the basin’s radius by 2 km, by how many square kilometers does the area decrease?", "### Calculating the Original Radius", "The area of a circle is given by the formula:", "[\nA = \pi r^2\n]", "Given $ A = 100\pi , \ ext{km}^2 $, we solve for $ r $:", "[\n\pi r^2 = 100\pi\n]", "Divide both sides by $ \pi $:", "[\nr^2 = 100 \quad \Rightarrow \quad r = 10 , \ ext{km}\n]", "So, the original radius is 10 km.", "### Determining the New Radius After Sedimentation", "The radius decreases by 2 km:", "[\nr_{\ ext{new}} = 10 - 2 = 8 , \ ext{km}\n]", "### Calculating the New Basin Area", "Using the area formula again with the reduced radius:", "[\nA_{\ ext{new}} = \pi (8)^2 = \pi \ imes 64 = 64\pi , \ ext{km}^2\n]", "### Calculating the Area Decrease", "The reduction in area is the difference between the original and new areas:", "[\n\Delta A = 100\pi - 64\pi = 36\pi , \ ext{km}^2\n]", "Since the question asks for the decrease in square kilometers, and $ \pi \approx 3.1416 $, we can compute numerically:", "[\n36\pi \approx 36 \ imes 3.1416 = 113.0976 , \ ext{km}^2\n]", "However, for mathematical precision in an official context, keeping $ \pi $ is preferred:", "[\n\Delta A = 36\pi , \ ext{km}^2\n]", "Thus, the drainage basin’s area decreases by $ 36\pi , \ ext{km}^2 $, equivalent to approximately 113.1 km² when calculated numerically.", "### Why This Matters for Hydrological Modeling", "Changes in radius—whether due to erosion, deposition, or human intervention—directly influence a basin’s water-holding capacity. A 2 km reduction represents a significant shrinkage that diminishes runoff storage and flood mitigation potential. Hydrologists incorporate such geometric changes into predictive models to assess water availability, flood risk, and long-term sustainability of watersheds affected by sedimentation and land-use changes.", "In summary, a 2 km reduction in radius from a circular drainage basin with an original area of $ 100\pi , \ ext{km}^2 $ results in a measurable area decrease of $ 36\pi , \ ext{km}^2 $, illustrating the measurable hydrological impacts of sedimentation in natural systems.", "---", "Keywords: hydrologist, circular drainage basin, area decrease, sedimentation, water storage, hydrological modeling, $ \pi $, $ 100\pi , \ ext{km}^2 $, 36π km², radius reduction"]