To find the time at which maximum height is reached, use the vertex formula $ t = -\fracb2a $ for the quadratic equation $ h(t) = -5t^2 + 20t + 10 $. - Malaeb
How to Find the Time at Which Maximum Height Is Reached Using the Vertex Formula
How to Find the Time at Which Maximum Height Is Reached Using the Vertex Formula
When analyzing the motion of an object under gravity, such as a ball thrown into the air, one key question is: At what time does the object reach its maximum height? For quadratic equations that model height over time, the answer lies in finding the vertex of the parabola represented by the equation. This allows us to determine the exact moment of peak elevation without graphing the function.
Understanding the Quadratic Model
Understanding the Context
In physics and mathematics, projectile motion is often described by a quadratic equation of the form:
$$
h(t) = at^2 + bt + c
$$
where:
- $ h(t) $ is the height at time $ t $,
- $ a $, $ b $, and $ c $ are constants,
- $ a < 0 $ ensures the parabola opens downward, meaning there is a maximum point.
In our case, the height function is:
Image Gallery
Key Insights
$$
h(t) = -5t^2 + 20t + 10
$$
Here, $ a = -5 $, $ b = 20 $, and $ c = 10 $. Since $ a $ is negative, the parabola opens downward, so the vertex represents the peak height and the corresponding time.
Using the Vertex Formula
To find the time $ t $ at which the maximum height is reached, use the vertex formula:
$$
t = - rac{b}{2a}
$$
🔗 Related Articles You Might Like:
📰 tom siwa 📰 did ozzy osbourne die by assisted death 📰 kaleycuoconude 📰 Windows 11 Build 25 Just Droppedthese Updates Will Transform Your Experience 4664463 📰 Games Releasing In 2024 6234591 📰 Joint Distribution 2267766 📰 La Galaxy Vs Cruz Azul Timeline 9537532 📰 Givenchy Shark Boots This Shocking Style Is Ruining Saving Room Tryoutsshop Now 4308475 📰 Verizon Cell Outage 3950155 📰 My Verizon Prepaid Refill 582490 📰 Shadow Returns Like Never Before Dive Into Super Shadow Sonic 3 Now 8781213 📰 Knuckles Sonic 9550962 📰 Aaron Rodgers Girlfriend 1193920 📰 Nike Stock Is About To Skyrocket Dont Miss Out On This Limited Opportunity 8233822 📰 This Is The Zodiac Speaking 7504249 📰 Only Fans Descargar App 7851657 📰 Lacey Lane Aew Signing 8858630 📰 The Elite Actor Who Bring Albus Dumbledore To Life You Need To See This Before It Goes Viral 5687309Final Thoughts
Substituting $ a = -5 $ and $ b = 20 $:
$$
t = - rac{20}{2(-5)} = - rac{20}{-10} = 2
$$
Thus, the maximum height is achieved at $ t = 2 $ seconds.
Why This Works
The vertex formula derives from completing the square or using calculus, both confirming that the axis of symmetry of the parabola lies at $ t = - rac{b}{2a} $. This time corresponds to the peak of the motion — exactly when the upward velocity becomes zero and the object begins descending.
Real-World Application
Imagine throwing a ball straight upward. Even without graphic tools, using $ h(t) = -5t^2 + 20t + 10 $, you instantly know the ball peaks at $ t = 2 $ seconds — critical for catching it at its highest point or assessing impact timing.
Summary:
To find the time of maximum height in a quadratic motion model, apply $ t = - rac{b}{2a} $. For $ h(t) = -5t^2 + 20t + 10 $, this yields $ t = 2 $. This method simplifies vertical motion analysis and supports physics-based problem solving.
