{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t²  where t is measured in seconds after the hit.


e. At what time is the ball at a height of 10 ft on the way down? 

Verified step by step guidance

1. Step 1: Set up the equation for the height of the ball. We are given that the height function is \( h(t) = 64t - 16t^2 \). We need to find the time \( t \) when the height \( h(t) = 10 \) feet.
2. Step 2: Substitute 10 for \( h(t) \) in the equation: \( 10 = 64t - 16t^2 \).
3. Step 3: Rearrange the equation to form a standard quadratic equation: \( 16t^2 - 64t + 10 = 0 \).
4. Step 4: Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( t \), where \( a = 16 \), \( b = -64 \), and \( c = 10 \).
5. Step 5: Calculate the discriminant \( b^2 - 4ac \) and then find the two possible values for \( t \). Choose the value of \( t \) that corresponds to the ball being on its way down.

Key Concepts

Quadratic Functions

The height function h(t) = 64t - 16t² is a quadratic function, which is a polynomial of degree two. Quadratic functions graph as parabolas and can model various physical phenomena, such as projectile motion. Understanding the properties of parabolas, including their vertex and intercepts, is essential for analyzing the height of the baseball over time.

Solving Quadratic Equations

To find the time when the baseball reaches a height of 10 ft, we need to solve the equation 64t - 16t² = 10. This involves rearranging the equation into standard form and applying methods such as factoring, completing the square, or using the quadratic formula. Mastery of these techniques is crucial for determining specific values of t in quadratic scenarios.

Projectile Motion

The scenario describes projectile motion, where an object is thrown vertically and affected by gravity. The height of the object over time can be modeled using a quadratic equation, where the initial velocity and gravitational acceleration influence the trajectory. Understanding the principles of projectile motion helps in interpreting the behavior of the baseball as it rises and falls.