A company's profit function is modeled by \(P(x) = -5x^2 + 150x - 1000\), where \(x\) is the number of units sold. Find the number of units that maximizes profit.
["# Maximizing Profit: How to Determine the Optimal Number of Units to Sell", "In business, profit maximization is a core objective. For one company, the profit function is modeled by:\n[\nP(x) = -5x^2 + 150x - 1000\n]\nwhere ( x ) represents the number of units sold. Understanding the optimal production level helps maximize revenue while minimizing costs and risks. In this article, we’ll analyze this quadratic profit function to find the number of units that maximizes profit.", "## The Nature of the Profit Function", "The given profit function is a quadratic equation in the standard form:\n[\nP(x) = ax^2 + bx + c\n]\nHere, ( a = -5 ), ( b = 150 ), and ( c = -1000 ). Since ( a < 0 ), the parabola opens downward, meaning the vertex represents the maximum value—exactly the point where profit is maximized.", "## Finding the Vertex of the Parabola", "For any quadratic function of the form ( ax^2 + bx + c ), the ( x )-value at the vertex (which gives the point of maximum or minimum) is found using the formula:\n[\nx = -\frac{b}{2a}\n]\nSubstituting ( a = -5 ) and ( b = 150 ):\n[\nx = -\frac{150}{2(-5)} = -\frac{150}{-10} = 15\n]", "## Interpretation and Conclusion", "The number of units that maximizes profit is ( x = 15 ). This means selling 15 units yields the highest profit according to the function. At this point, the rate of revenue increase balances with rising costs, achieving optimal financial performance.", "To verify, substituting ( x = 15 ) into the profit function:\n[\nP(15) = -5(15)^2 + 150(15) - 1000 = -1125 + 2250 - 1000 = 125\n]\nProfit reaches its peak at $125 when 15 units are sold.", "In summary, applying the vertex formula efficiently identifies the production level that maximizes profit—critical insight for strategic decision-making.", "---\nFOCUS: Maximize profit by producing 15 units.\nTECHNIQUE: Use the vertex formula ( x = -\frac{b}{2a} ) on quadratic functions.\nBENEFIT: Precise optimization supports optimal revenue generation."]
