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 $. - RTA
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:
📰 substandard meaning 📰 fiden 📰 saja meaning korean 📰 Marriott Grand Residence Club Lake Tahoe 1224951 📰 Super Affordable Thermal Paste Near Youclick To Find Deals Before Stock Runs Out 1544653 📰 Anet Stock Insider Tips How To Ride This Explosive Growth Before It Peaks 5620256 📰 You Wont Believe What Lorraine Toussaint Revealed In Her Latest Tv Star Appearances 9175471 📰 What Does Pink Mean 7210777 📰 Equation Of Acetic Acid 360427 📰 Lightning Mcqueens Ride More Than Just Fire And Furyheres Why Its Lightnings Throne 9100935 📰 What Are Erp Systems The Simple Explanation That Could Transform Your Business 5607662 📰 Www Verizonwireless Fraud 8038819 📰 I Built A Luxurious Chair And A Half Oasisyou Wont Believe How Spacious It Is 6565804 📰 5 Oz Pounds 6278830 📰 You Wont Believe What Happens When You Connectit Changed Everything Forever 1773433 📰 Mortgage Loans For First Time Home Buyers 5070619 📰 The Girl Who Loved Tom Gordon 4614909 📰 Perfect 32 Bit Iso For Windows 7 Heres Everything You Need In 2024 5313949Final 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.
