https://file.aiquickdraw.com/mathbot/file/1727073304591y483bjo1.jpg

We are given a height function of a rocket's motion:

\[
h(t) = -4.9t^2 + 202t + 235
\]

To find the maximum height, we first need to determine the time \( t \) at which the height is maximized. This is a quadratic equation in the form \( h(t) = at^2 + bt + c \), where \( a = -4.9 \), \( b = 202 \), and \( c = 235 \). Since the coefficient \( a \) is negative, the parabola opens downwards, meaning the vertex represents the maximum point.

The formula to find the time at which the maximum height occurs (the vertex of the parabola) is:

\[
t = \frac{-b}{2a}
\]

Let me calculate this and then find the corresponding height \( h(t) \) for the maximum value.The rocket reaches its maximum height at \( t = 20.6 \) seconds. At that time, the maximum height is approximately 2316.8 meters above sea level.

Would you like more details on the steps or any clarification?

Here are 5 related questions to consider:
1. How would the maximum height change if the initial velocity term (202) were different?
2. What role does gravity (the \(-4.9\)) play in determining the rocket's height?
3. What would the rocket's height be at any other specific time?
4. Can you calculate the time it takes for the rocket to hit the ground after launch?
5. How does the initial height (235) affect the overall flight?

**Tip:** The vertex formula for quadratic equations always gives the maximum or minimum point for any parabolic motion, depending on the sign of \(a\).

A rocket is launched in the air. Its height (in meters above sea level) as a function of time (in seconds) is given by h(t) = -4.9t^2 + 202t + 235. Find the maximum height the rocket attains.

Learn how to find the maximum height of a rocket launched using the quadratic equation h(t) = -4.9t^2 + 202t + 235. This problem involves key algebraic concepts, including maximization and the use of the vertex formula.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation