Since \( a < 0 \), the parabola opens downward, so the maximum occurs at the vertex: \( t = -\frac{b}{2a} = -\frac{150}{2(-5)} = 15 \) seconds.

["Title: Finding the Maximum of a Downward-Opening Parabola: The Vertex Formula Explained", "When modeling real-world phenomena using quadratic functions, one of the most important tasks is identifying where the maximum or minimum value occurs. For parabolas defined by quadratic equations in the form ( f(t) = at^2 + bt + c ), the direction the parabola opens depends on the coefficient ( a )—and since this problem involves ( a < 0 ), the parabola opens downward, guaranteeing the presence of a maximum point.", "Given a quadratic equation such as ( f(t) = -5t^2 + 150t + c ) (where ( a = -5 )), the negative value of ( a ) confirms the parabola opens downward, shaping a "mountain" shape that reaches a peak at its vertex. Understanding how to compute the vertex is essential for optimization problems across physics, engineering, economics, and beyond.", "### Key Formula: The Vertex of a Parabola", "The vertex of a parabola in standard form lies at ( t = -\frac{b}{2a} ). This formula derives from calculus or symmetry—essentially finding the axis of symmetry which bisects the parabola. For any quadratic equation with ( a < 0 ), this formula guarantees the maximum occurs precisely at this ( t )-value.", "### Applying the Values", "Suppose we analyze a quadratic model such as:\n[\nf(t) = -5t^2 + 150t + c\n]\nHere, ( a = -5 ) and ( b = 150 ). Plugging into the vertex formula:\n[\nt = -\frac{b}{2a} = -\frac{150}{2(-5)} = -\frac{150}{-10} = 15\n]", "Thus, the maximum value occurs at ( t = 15 ) seconds. This critical insight allows quick identification of peak performance time, peak height, or optimal decision points in practical applications.", "### Why This Matters", "In motion analysis, for example, a projectile’s vertical displacement modeled by such a quadratic reaches its maximum height exactly at ( t = 15 ), helping engineers and physicists predict optimal launch timing or target impact points. In business, similar models can represent profit columns or cost surfaces, where the downward opening indicates a single point of peak efficiency.", "### Summary", "- Since ( a < 0 ), the parabola opens downward, confirming a maximum value exists.\n- The vertex occurs at ( t = -\frac{b}{2a} ), a standard tool in quadratic analysis.\n- In this case, with ( a = -5 ) and ( b = 150 ), the maximum occurs at ( t = 15 ) seconds.", "Understanding this calculation empowers accurate interpretation and decision-making based on quadratic models in both academic and real-world contexts.", "---", "Keywords: parabola opens downward, maximum of quadratic, vertex formula, ( t = -b/(2a) ), optimization, quadratic functions, physics modeling, calculus in motion, real-world applications."]