A company's profit is modeled by \(P(x) = -2x^2 + 40x - 150\), where x is the number of units sold. What is the maximum profit?
["Title: How to Maximize Profit: Analyzing the Quadratic Model ( P(x) = -2x^2 + 40x - 150 )", "Understanding how profit functions are modeled mathematically is essential for businesses aiming to maximize profits efficiently. One such model used by companies to forecast profit is the quadratic function:", "[\nP(x) = -2x^2 + 40x - 150\n]", "where ( x ) represents the number of units sold.", "### Why is the Profit Function Quadratic?", "This downward-opening parabola represents the realistic economic idea that profit initially increases with sales but eventually declines due to costs, competition, or market saturation—common types of marginal returns. The highest point of the curve corresponds to the maximum profit, a critical value for strategic planning.", "---", "### How to Find the Maximum Profit", "The profit function ( P(x) = -2x^2 + 40x - 150 ) is a quadratic equation in standard form ( ax^2 + bx + c ). Because the coefficient of ( x^2 ) (i.e., ( a = -2 )) is negative, the parabola opens downward, and the vertex gives the maximum point.", "For a quadratic function ( ax^2 + bx + c ), the x-coordinate of the vertex is given by:", "[\nx = -\frac{b}{2a}\n]", "Substitute ( a = -2 ) and ( b = 40 ):", "[\nx = -\frac{40}{2(-2)} = -\frac{40}{-4} = 10\n]", "So, the number of units that maximizes profit is ( x = 10 ).", "---", "### Calculate Maximum Profit", "Now substitute ( x = 10 ) into the profit function:", "[\nP(10) = -2(10)^2 + 40(10) - 150\n]", "[\nP(10) = -2(100) + 400 - 150\n]", "[\nP(10) = -200 + 400 - 150 = 50\n]", "---", "### Conclusion: Maximum Profit is $50", "The maximum profit the company can achieve is $50, attained when 10 units are sold.", "This model helps businesses identify optimal production levels and inform strategic decisions such as pricing, marketing spend, and inventory management. By leveraging mathematical modeling, companies can shift from guesswork to data-driven profit maximization.", "---", "Key Takeaways:\n- The vertex of a quadratic profit function gives the maximum output.\n- For ( P(x) = ax^2 + bx + c ), use ( x = -\frac{b}{2a} ).\n- Plug the optimal ( x ) back into the equation to find maximum profit.\n- Practical application: Maximize revenue with clear, quantifiable targets.", "---", "Optimize your business strategy today—start modeling profit with a quadratic function!\nFor more insights on financial modeling, refer to standard business economics frameworks."]
