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:

📰 kingdom hearts 4 release date 📰 kingdom hearts characters 📰 kingdom hearts coded 📰 You Wont Believe The Hidden Mistake In Missing Letter Crossword Puzzles 464590 📰 Darksiders Ii 1255213 📰 Horse Council Reveals Shocking Secret No One Discusses 8392534 📰 Battlefield One On Steam 7576884 📰 The Escapists Unleashed How These Players Lost Themselves In Virtual Worlds Forever 6958630 📰 Www Mynetworksettings Com Verizon 5557855 📰 Highest Yielding Savings Accounts 3679949 📰 You Wish You Knew How Toastul Changes Everything 3480964 📰 These 70S Hairstyles Changed Mens Fashion Foreverwatch How They Dominated The Decade 5325507 📰 Best Air Filters For Home 6704640 📰 Tv Las Vegas Show 7277428 📰 Sign In For Act 9978797 📰 From Chaos To Clarity How Canvas School App Is Revolutionizing Class Management 3380874 📰 Le Premier Terme Est 5 Le Dernier 95 La Raison 5 Rsolvez 5 N 1 Times 5 95 Rightarrow N 19 5414364 📰 Actors From The Internship 3142767

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.