Sharon kicks a ball from the ground into the air with an upward velocity of 64 feet per second. The function [tex]$h=-16 t^2+64 t$[/tex] models the height [tex]$h$[/tex], in feet, of the ball at time [tex]$t$[/tex], in seconds. When will the ball reach the ground again?
A. 3 seconds after the ball is thrown
B. 4 seconds after the ball is thrown
C. 2 seconds after the ball is thrown
D. 1 second after the ball is thrown
Answer (1)
Set the height function equal to zero: − 16 t 2 + 64 t = 0 .
Factor the equation: − 16 t ( t − 4 ) = 0 .
Solve for t : t = 0 or t = 4 .
The ball reaches the ground again at t = 4 seconds: 4 seconds .
Explanation
Understanding the Problem We are given the height function h = − 16 t 2 + 64 t , which models the height of the ball at time t . We want to find the time when the ball reaches the ground again. This means we need to find the value of t when h = 0 .
Setting up the Equation To find when the ball reaches the ground, we set the height function equal to zero: − 16 t 2 + 64 t = 0
Factoring the Equation Now, we factor out the common factor of − 16 t from the equation: − 16 t ( t − 4 ) = 0
Solving for t Next, we set each factor equal to zero and solve for t :
− 16 t = 0 or t − 4 = 0 Solving these equations gives us: t = 0 or t = 4
Finding the Time The solution t = 0 represents the initial time when the ball is kicked from the ground. The solution t = 4 represents the time when the ball reaches the ground again. Therefore, the ball will reach the ground again 4 seconds after it is thrown.
Final Answer The ball will reach the ground again after 4 seconds.
Examples
Understanding projectile motion, like the path of Sharon's ball, is crucial in many real-world applications. For example, engineers use these principles to design trajectories for rockets and satellites. Athletes in sports like basketball and soccer also intuitively use these concepts to aim their shots. By understanding the parabolic path described by the equation h = − 16 t 2 + 64 t , we can predict how long an object will remain airborne and how far it will travel, optimizing performance and safety in various scenarios.
