A company’s profit in thousands of dollars is modeled by \( P(x) = -2x^2 + 20x - 30 \), where \( x \) is the number of units sold in hundreds. What is the maximum profit, and how many units maximize it?

["Maximizing Profit: How a Company’s Earnings Peak at a Strategic Production Level", "In business, understanding profit dynamics is crucial for optimizing performance. A recent earnings model for a manufacturing company is given by the quadratic profit function:\n[\nP(x) = -2x^2 + 20x - 30\n]\nwhere ( x ) represents the number of units sold in hundreds of units, and ( P(x) ) is the profit in thousands of dollars.", "This article explores how to determine the maximum profit and the production level that achieves it—key insights for strategic decision-making.", "---", "### Understanding the Profit Function", "The function ( P(x) = -2x^2 + 20x - 30 ) is a quadratic equation. Because the coefficient of ( x^2 ) is negative ((-2)), the parabola opens downward, meaning it has a single maximum point at its vertex.", "This shape reflects real-world scenarios where profit initially rises with increased sales but eventually declines due to rising costs, capacity limits, or market saturation.", "---", "### Finding the Maximum Profit and Optimal Units Sold", "For any quadratic function ( ax^2 + bx + c ), the vertex—where the maximum (or minimum) occurs—lies at:\n[\nx = -\frac{b}{2a}\n]", "Here, ( a = -2 ), ( b = 20 ), so:\n[\nx = -\frac{20}{2 \ imes (-2)} = -\frac{20}{-4} = 5\n]", "Since ( x ) represents units in hundreds, selling 5 units corresponds to:\n[\n5 \ imes 100 = 500 \ ext{ physical units}\n]", "Substitute ( x = 5 ) back into ( P(x) ) to find the maximum profit:\n[\nP(5) = -2(5)^2 + 20(5) - 30 = -2(25) + 100 - 30 = -50 + 100 - 30 = 20\n]", "Thus, the maximum profit is $20,000.", "---", "### Interpretation and Business Insight", "- Maximum Profit: The highest achievable profit is $20,000 per 100-unit increment, or $200,000 on a thousand-dollar scale—though correctly interpreted as $20,000 due to model units.\n- Optimal Production: To maximize earnings, the company should aim to sell 500 units per production cycle, balancing demand and cost efficiency.", "---", "### Final Thoughts", "By analyzing the vertex of the profit function, the company identifies a clear, data-driven target: producing and selling 500 units maximizes profitability, ensuring sustainable growth. Using mathematical modeling, leaders can make informed decisions to optimize operations and financial performance.", "---", "Key Takeaways:\n- Maximum profit occurs at ( x = 5 ) (500 units).\n- Maximum profit value: $20,000.\n- Use vertex form of quadratics for quick optimization.", "---", "Modeled in hours, profits in thousands—this balance of math and machine defines modern business success."]