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 $. - AIKO, infinite ways to autonomy.
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:
📰 You Wont Believe the Simple Contribution Limits 2025 That Could Change Your Donations Forever! 📰 Unlock the Hidden Secret: Simple Contribution Limits 2025 You Must Know! 📰 This 2025 Rule on Contribution Limits Is Shocking—Start Complying Before Its Too Late! 📰 You Wont Believe Where Ellen Is Nowher Secret Life Is Wild Compulsive Click Load 2224039 📰 What Time Is It With Daylight Savings 2443235 📰 The Untold Truth About Myinstants That Will Change How You See Every Moment 9322247 📰 Jackson Kim 304675 📰 The Shocking Scale Of Destruction Discover The True Blast Radius Of A Nuclear Bomb 6336383 📰 Gheit Is Increasingly Viewed As A Core Midfielder At Villa With Interest From European Clubs Evident Its Retention Reflects Both His Progress And The Clubs Belief In Elevating Home Talent He Aims To Secure A Regular Place And Grow Into A Leader For Romanias Senior Team 2055663 📰 You Wont Believe What Makes A Moto Racer Unstoppable 6147037 📰 George Foreman Sells Choice Home Warranty That Saves Georges Futureand Yoursbefore You Sleep 8756963 📰 Cumin On Face Hack The Natural Glow Youre Cravingheres How 5889237 📰 Tattoo Ideas For Men 2293343 📰 Online Shooter Games Free 4393062 📰 Love Fighting Games Try These Free Online Fighting Games You Can Play Nowno Pay 1930342 📰 2025 Direct Deposit Schedule 1565460 📰 Listeria Outbreak California 7461112 📰 La Super Rica Taqueria 8875854Final 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.
