Question:** A chemical engineer is modeling biofuel yield based on reactant concentrations. If the yield \(Y\) is given by \(Y = 4x^2 - 12x + 9\), compute the minimum possible yield and the value of \(x\) at which it occurs.
["Modeling Biofuel Yield: Finding the Minimum Efficiency Using Quadratic Analysis", "In the pursuit of sustainable energy, chemical engineers frequently model the efficiency of biofuel production through mathematical functions. One common scenario involves optimizing reactant concentration to maximize yield—though in this case, we focus on minimizing yield based on a quadratic expression.", "Suppose the yield (Y) of a biofuel process is modeled by the function:\n[\nY = 4x^2 - 12x + 9\n]\nwhere (x) represents the reactant concentration. Understanding where this yield achieves its minimum is critical for process optimization and resource efficiency.", "---", "### Finding the Minimum Yield of a Quadratic Function", "Quadratic functions of the form (Y = ax^2 + bx + c) achieve their extreme values—minimum or maximum—at the vertex. Since the coefficient (a = 4) is positive, the parabola opens upwards, confirming the vertex is a minimum point.", "The (x)-coordinate of the vertex is given by:\n[\nx = -\frac{b}{2a}\n]\nSubstituting (a = 4) and (b = -12):\n[\nx = -\frac{-12}{2 \ imes 4} = \frac{12}{8} = 1.5\n]", "Now substitute (x = 1.5) back into the yield equation to find the minimum yield:\n[\nY = 4(1.5)^2 - 12(1.5) + 9\n]\n[\nY = 4(2.25) - 18 + 9 = 9 - 18 + 9 = 0\n]", "---", "### Conclusion", "The minimum possible yield is (Y = 0), and this occurs when the reactant concentration is (x = 1.5). While zero yield may seem undesirable, this result implies that at optimal reactant levels modeled here, the process achieves zero biofuel production—possibly due to inhibitory effects, near-equilibrium conditions, or process inefficiencies at that point. Recognizing such critical points helps engineers adjust conditions to avoid inefficiencies and improve overall biofuel output.", "This mathematical modeling demonstrates how chemical engineers leverage algebra and calculus to refine production parameters, ultimately enhancing sustainability and profitability in biofuel manufacturing.", "---", "Keywords: biofuel yield, chemical engineer, quadratic function modeling, reactant concentration, minimum yield calculation, vertex of parabola, yield optimization."]
