Question: A hydrologist designs a trapezoidal reservoir with an inscribed circle. If the sum of the non-parallel sides is $ 20 $ meters and the radius of the circle is $ 5 $ meters, what is the area of the reservoir?
["A hydrologist designs a trapezoidal reservoir with an inscribed circle. If the sum of the non-parallel sides is 20 meters and the radius of the circle is 5 meters, what is the area of the reservoir?", "Why This Question is Rising in US Water Infrastructure Research \nAdvances in sustainable water storage are gaining momentum across the United States, driven by climate resilience and aging infrastructure. The design of efficient, circular-integrated reservoirs—especially those with inscribed geometric constraints—has sparked interest among civil engineers and environmental planners. This particular question reflects real-world focus: optimizing trapezoidal geometries where hydrological function converges with geometric precision, supported by proven principles like an inscribed circle. As water scarcity and flood management evolve as national priorities, such technical inquiries highlight the fusion of classical geometry and modern civil engineering.", "The Science Behind Trapezoidal Reservoirs with Inscribed Circles \nIn trapezoidal reservoirs designed to hold water efficiently, stability and structural integrity are paramount. When an inscribed circle fits perfectly within the trapezoid—a shape with one pair of parallel sides and non-parallel sides meeting at precise angles—its radius becomes a key geometric indicator. For a trapezoid with an inscribed circle, a well-established rule holds: the sum of the lengths of the two non-parallel sides equals the sum of the two parallel sides. Here, the sum of the non-parallel sides is given as 20 meters, so the sum of the parallel sides must also be 20 meters.", "With the diameter of the inscribed circle equaling twice the radius—here, 10 meters—the diameter matches the shorter base projection across the trapezoid. This alignment reveals geometry that engineers use to predict load distribution, water retention, and flow dynamics. Understanding this relationship strengthens planning for sustainable water storage systems across diverse US landscapes, from arid plains to flood-prone regions.", "How Much Area Does This Reservoir Hold? \nTo calculate the area, recall the formula for the area of a trapezoid: \n\[\n\ ext{Area} = \frac{1}{2} \ imes (a + b) \ imes h\n\] \nwhere \(a\) and \(b\) are the parallel sides, and \(h\) is the height. In this case, \(a + b = 20\) meters, and the inscribed circle’s diameter (10 meters) confirms the height \(h = 10\) meters is consistent with the radius and circulation mechanics.", "Multiply: \n\[\n\ ext{Area} = \frac{1}{2} \ imes 20 \ imes 10 = 100 \ ext{ square meters}\n\] \nThus, the reservoir spans 100 m²—sufficient for small"]
