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 $. - Midis
The Full Solution to Finding the Maximum Value of the Quadratic Function $ p(t) = -t^2 + 14t + 30 $
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.
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:
📰 2) What Does HG Mean? The Surprising Truth Women Will Never Tell You 📰 3) HG Code Decoded: What This Acronym Actually REVEALS About Mysterious Safety Terms 📰 4) The Hidden Meaning Behind HG—Every Internet User Needs to Know Now 📰 You Wont Believe What Happens At The Next Open Mic Night Around You 📰 You Wont Believe What Happens If You Violate The No Touch Law 📰 You Wont Believe What Happens In Paulding County 📰 You Wont Believe What Happens In The Ninja Kidz Action Park 📰 You Wont Believe What Happens Inside Myclubonlines Hidden World 📰 You Wont Believe What Happens When Brothers Celebrate National Mens Day 📰 You Wont Believe What Happens When Elements Clash Charges 📰 You Wont Believe What Happens When Heat Turns Your Body Inside Outskyrocket Safer Than You Think 📰 You Wont Believe What Happens When Nad Goes Directly Into Your System 📰 You Wont Believe What Happens When Nanas Brew Green Tea For A Hour 📰 You Wont Believe What Happens When Nations Compete In Sports 📰 You Wont Believe What Happens When Necco Wafers Touch Textures 📰 You Wont Believe What Happens When Ned Rig Takes Control 📰 You Wont Believe What Happens When Nine Cups Reveal Your Fate 📰 You Wont Believe What Happens When Offsides Ruin The GameFinal 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.
