Math Problem Statement
Solution
The problem gives us the height of a baseball, where is the time in seconds after it is hit. The height function is:
1. Time to Reach Maximum Height:
To find the time it takes to reach the maximum height, we can use the fact that the maximum of a quadratic function occurs at:
Here, and . Plugging in these values:
So, it takes 3.5 seconds for the baseball to reach its maximum height.
2. Maximum Height:
Now, substitute into the height function :
So, the maximum height reached by the baseball is 61.0 meters (rounded to one decimal place).
Would you like further details or have any other questions about the steps?
Here are 5 related questions to consider:
- How would the problem change if the initial velocity was different?
- What happens to the maximum height if we change the gravitational constant?
- Can we solve for the time when the baseball returns to the ground?
- How can we model the trajectory of the baseball with air resistance included?
- How would you determine the velocity of the baseball at its maximum height?
Tip: The vertex form of a quadratic equation is helpful for quickly determining maxima or minima in projectile motion problems!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Projectile Motion
Vertex of a Parabola
Formulas
Quadratic formula for maximum height: t = -b / (2a)
Height function: h(t) = -4.9t^2 + 34.3t + 1
Theorems
Quadratic Maximum Theorem
Suitable Grade Level
Grades 9-12
Related Recommendation
Find the Maximum Height of a Ball Using Quadratic Equations
Solving Quadratic Equations for Maximum Height and Horizontal Distance
Finding Maximum Height Time for Baseball Flight Modeled by Quadratic Function
Find the Maximum Height and Horizontal Distance of a Thrown Ball Using a Quadratic Equation
Projectile Motion: Solving Quadratic Functions for Baseball Pop-Ups