The first equation in the system models the height, $h$, of a falling volleyball as a function of time, $t$. The second equation models the height, $h$, of the hands of a player jumping up to spike the ball as a function of time, $t$. Which statement describes the situation modeled by this system?
$\left\{\begin{array}{l}
h=14-16 t^2 \\
h=7+24 t-16 t^2
\end{array}\right.$
A. The volleyball is 7 feet above the ground at the instant the player begins her jump.
B. The volleyball is 14 feet above the ground at the instant the player begins her jump.
C. The volleyball is 16 feet above the ground at the instant the player begins her jump.
D. The volleyball is 24 feet above the ground at the instant the player begins her jump.
Answer (1)
Substitute t = 0 into the volleyball height equation.
Calculate h = 14 − 16 ( 0 ) 2 .
Simplify to find the initial height.
The volleyball's initial height is 14 feet.
Explanation
Understanding the Problem We are given two equations:
The height of the volleyball: h = 14 − 16 t 2
The height of the player's hands: h = 7 + 24 t − 16 t 2
We want to find the initial height of the volleyball at the instant the player begins her jump. The player begins her jump at time t = 0 . We need to substitute t = 0 into the equation for the height of the volleyball to find its initial height.
Calculating the Initial Height Substitute t = 0 into the equation for the height of the volleyball:
h = 14 − 16 ( 0 ) 2
h = 14 − 16 ( 0 )
h = 14 − 0
h = 14
Final Answer The initial height of the volleyball is 14 feet. Therefore, the volleyball is 14 feet above the ground at the instant the player begins her jump.
Examples
Understanding the initial conditions of projectile motion is crucial in sports like volleyball. For instance, knowing the initial height of the volleyball ( h = 14 − 16 t 2 ) helps players anticipate its trajectory. Similarly, the player's jump ( h = 7 + 24 t − 16 t 2 ) can be analyzed to optimize their timing and reach. By analyzing these equations, coaches and players can develop strategies to improve performance, such as determining the best moment to jump or the ideal height to set the ball for a spike. This blend of math and sports enhances precision and decision-making on the court.
