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:

📰 Average Apr Financing Car 📰 Best Small Business Savings Account 📰 Roth Ira Guide 📰 How The Oompa Loompa Costume Dominated Costume Awards Proven By Science 9919848 📰 Asian Cuisine Dessert 1434277 📰 Credit Union Cd Rates 9193791 📰 Big Ass Asian Beauty Power How These Iconic Stars Dominated Every Room Dont Miss Out 2968731 📰 Park Pelican 7995923 📰 What Is A Tsundere The Shocking Truth About Why Theyre Overly Angry And Then So Passive 4356992 📰 Palladium Etf Surge This Rare Metals Etf Just Shocked The Marketdont Miss Out 6399963 📰 Aca Insurance 3202681 📰 Spend Zero Money Top Free Online Puzzle Games For Windows You Need To Try 5468532 📰 Blitz The League Only Top Gun Teams Survive The Rapid Fire Showdown 7855672 📰 Civic Credit Union Betrayal What Members Dont Want You To Know 504319 📰 Youll Never Believe What Happened The Next Time The N Train Cracks A Stop Sign 2217978 📰 How Much Is It For A Apple Watch 6575892 📰 Wells Fargo Fraud 5444511 📰 Diego Diego Finally Exposes His Truthyou Wont Believe Who He Really Is 2170180

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.