Question: A sustainable farming algorithm predicts water usage $ W = 5x^2 - 20x + 45 $. Find the smallest $ x $ that minimizes $ W $.
["Sequence in Sustainable Farming: How a Sustainable Farming Algorithm Predicts Optimal Water Usage", "In modern sustainable agriculture, precise resource management is essential. One innovative approach uses mathematical algorithms to optimize water usage, a critical factor in preserving limited freshwater resources. A powerful tool in this domain is a quadratic model predicting water consumption:\n[ W = 5x^2 - 20x + 45 ]\nwhere $ x $ represents an adjustable farming parameter such as soil moisture level, irrigation intensity, or crop density. Understanding how to minimize this function leads to sustainable and efficient farming practices.", "### Why Optimizing Water Usage Matters\nWater scarcity impacts food security worldwide, making efficient irrigation vital. By modeling water usage mathematically, farmers and agricultural scientists can fine-tune inputs—reducing waste while maintaining crop yields. A key insight lies in finding the value of $ x $ that minimizes water consumption, reflecting the most sustainable operational balance.", "---", "### Understanding the Equation: $ W = 5x^2 - 20x + 45 $", "The given formula follows the standard quadratic form:\n[ W = ax^2 + bx + c ]\nWith coefficients:\n- $ a = 5 $\n- $ b = -20 $\n- $ c = 45 $", "Since $ a > 0 $, the parabola opens upward, indicating a single minimum point—the vertex. This vertex is the smallest value of $ W $, achievable at an optimal $ x $.", "---", "### Finding the Smallest $ x $ That Minimizes Water Usage", "To determine the vertex, use the vertex formula:\n[\nx = -\frac{b}{2a}\n]", "Plug in $ a = 5 $ and $ b = -20 $:\n[\nx = -\frac{-20}{2 \ imes 5} = \frac{20}{10} = 2\n]", "Thus, the smallest $ x $ value minimizing $ W $ is $ x = 2 $.", "At $ x = 2 $:\n[\nW = 5(2)^2 - 20(2) + 45 = 20 - 40 + 45 = 25\n]", "This confirms $ W = 25 $ is the minimal water usage under this model.", "---", "### Applying the Algorithm in Real Farming", "Farmers implement this algorithm by adjusting $ x $—such as irrigation timing or flow rate—based on soil sensors and environmental data. By targeting $ x = 2 $, they achieve minimal water usage without compromising crop health, improving sustainability metrics like water efficiency and crop productivity.", "---", "### Conclusion", "Sustainable farming algorithms like the one using $ W = 5x^2 - 20x + 45 $ demonstrate how mathematical modeling drives resource conservation. By identifying $ x = 2 $ as the optimal input that minimizes water usage, stakeholders gain actionable insight for eco-friendly farming. Leveraging such models empowers smarter decisions, helping secure water resources for future generations.", "Stay tuned to advancements in agricultural algorithms—where precision meets sustainability."]
