Solution: The quadratic $ p(t) = -t^2 + 14t + 30 $ opens downward. The vertex occurs at $ t = -\fracb2a = -\frac142(-1) = 7 $. Substituting $ t = 7 $, $ p(7) = -(7)^2 + 14(7) + 30 = -49 + 98 + 30 = 79 $.

The Full Solution to Finding the Maximum Value of the Quadratic Function $ p(t) = -t^2 + 14t + 30 $

Understanding key features of quadratic functions is essential in algebra and real-world applications such as optimization and curve modeling. A particularly common yet insightful example involves analyzing the quadratic function $ p(t) = -t^2 + 14t + 30 $, which opens downward due to its negative leading coefficient. This analysis reveals both the vertex — the point of maximum value — and the function’s peak output.

Identifying the Direction and Vertex of the Quadratic

Understanding the Context

The given quadratic $ p(t) = -t^2 + 14t + 30 $ is in standard form $ ax^2 + bx + c $, where $ a = -1 $, $ b = 14 $, and $ c = 30 $. Because $ a < 0 $, the parabola opens downward, meaning it has a single maximum point — the vertex.

The $ t $-coordinate of the vertex is found using the formula $ t = -rac{b}{2a} $. Substituting the values:

$$
t = -rac{14}{2(-1)} = -rac{14}{-2} = 7
$$

This value, $ t = 7 $, represents the hour or moment when the quantity modeled by $ p(t) $ reaches its maximum.

$Solution: The quadratic $ p(t) = -t^2 + 14t + 30 $ opens downward. The vertex occurs at $ t = -\frac{b}{2a} = -\frac{14}{2(-1)} = 7 $. Substituting $ t = 7 $, $ p(7) = -(7)^2 + 14(7) + 30 = -49 + 98 + 30 = 79 $.$

Key Insights

Calculating the Maximum Value

To find the actual maximum value of $ p(t) $, substitute $ t = 7 $ back into the original equation:

$$
p(7) = -(7)^2 + 14(7) + 30 = -49 + 98 + 30 = 79
$$

Thus, the maximum value of the function is $ 79 $ at $ t = 7 $. This tells us that when $ t = 7 $, the system achieves its peak performance — whether modeling height, revenue, distance, or any real-world behavior described by this quadratic.

Summary

🔗 Related Articles You Might Like:

📰 What No One is Saying About Sugar Mountain Assisted Living’s Mysterious Shutdown 📰 Dashed Hopes, Hidden Secrets—and the Dramatic Fall of Sugar Mountain Living 📰 Why Sugar Mountain Assisted Living Just Vanished Without Warning 📰 Report Reveals Connections Hint Dec 28 And It Raises Concerns 📰 Finally A Medicare Attorney Portal That Simplifies Your Benefitsclaim Your Share Today 6564283 📰 A Researcher Calculates That The Median Of A Dataset Of 8 Numbers Is 15 If A New Number 18 Is Added And The New Median Becomes 16 How Many Values Are Now Greater Than The Previous Median But Less Than The New Median 5518215 📰 Credit Card Debt Consolidation 📰 Viral Moment Cat Simulator Game And It Leaves Questions 📰 What Is Fabulous App 📰 When Will The Battle Pass End 8972615 📰 Best Ps1 Rpgs 📰 Watchos 26 1329435 📰 Best Buy Online Payment 6128382 📰 Descargar Downloader 📰 Transform Your Financial Power With Fidelity Backed Lines Of Credit Exclusive Offer Inside 4923271 📰 Bank Of America Open An Account Bonus 📰 Big Announcement Live Plane Tracking Map And The Internet Reacts 📰 Home Security Security Cameras 7451092

Final Thoughts

Function: $ p(t) = -t^2 + 14t + 30 $ opens downward ($ a < 0 $)
Vertex occurs at $ t = -rac{b}{2a} = 7 $
Maximum value is $ p(7) = 79 $

Knowing how to locate and compute the vertex is vital for solving optimization problems efficiently. Whether in physics, economics, or engineering, identifying such key points allows for precise modeling and informed decision-making — making the vertex a cornerstone of quadratic analysis.