Solution: The function $ h(t) = -5t^2 + 30t + 10 $ is a quadratic with a negative leading coefficient, so the maximum occurs at the vertex. The time $ t $ at which the maximum altitude occurs is:
["Understanding Quadratic Functions: Finding the Time of Maximum Altitude with $ h(t) = -5t^2 + 30t + 10 $", "When analyzing projectile motion using quadratic functions, nothing is more powerful than recognizing the role of the vertex. For equations of the form $ h(t) = -5t^2 + 30t + 10 $, this simple quadratic reveals when an object reaches its peak height — a crucial insight in physics, engineering, or mathematics.", "### Why the Negative Leading Coefficient Matters", "The general form of a quadratic is $ h(t) = at^2 + bt + c $, and the shape of its graph — a parabola — depends entirely on the sign of $ a $. Since our equation has $ a = -5 $, the negative coefficient tells us the parabola opens downward. This means the vertex represents the maximum point, not the minimum.", "Understanding this foundational concept ensures we set the right analysis — not minimizing altitude, but maximizing it at the vertex.", "### Calculating the Time at the Vertex", "For any quadratic $ h(t) = at^2 + bt + c $, the time $ t $ at which the vertex occurs is given by the formula:\n$$ t = -\frac{b}{2a} $$", "In our function, $ a = -5 $ and $ b = 30 $. Plugging in the values:", "$$\nt = -\frac{30}{2 \ imes (-5)} = -\frac{30}{-10} = 3\n$$", "Thus, the maximum altitude occurs at $ t = 3 $ seconds.", "### Conclusion", "The function $ h(t) = -5t^2 + 30t + 10 $ describes a downward-opening parabola due to its negative leading coefficient — confirming a maximum exists. By applying the vertex formula $ t = -\frac{b}{2a} $, we find the peak altitude happens at exactly 3 seconds. This principle applies broadly in modeling motion and optimization problems.", "Key takeaway: When modeling height over time with a quadratic, always use $ t = -\frac{b}{2a} $ to locate the maximum point — especially when the parabola opens downward."]
