The first equation in the system models the height in feet, $h$, of a falling baseball as a function of time, $t$. The second equation models the height in feet, $h$, of the glove of a player leaping up to catch the ball as a function of time, $t$. Which statement describes the situation modeled by this system?
$\left\{\begin{array}{l}
h=35-16 t^2 \\
h=6+18 t-16 t^2
\end{array}\right.$
A. The height of the baseball is 18 feet at the moment the player begins to leap.
B. The height of the baseball is 16 feet at the moment the player begins to leap.
C. The height of the baseball is 35 feet at the moment the player begins to leap.
D. The height of the baseball is 6 feet at the moment the player begins to leap.
Answer (1)
Substitute t = 0 into the equation h = 35 − 16 t 2 .
Calculate h = 35 − 16 ( 0 ) 2 = 35 .
The initial height of the baseball is 35 feet.
The correct statement is: The height of the baseball is 35 feet at the moment the player begins to leap. 35
Explanation
Understanding the Problem We are given a system of equations that model the height of a falling baseball and the height of a player's glove as a function of time. We need to determine the initial height of the baseball, which corresponds to the height at time t = 0 .
Finding the Initial Height The equation for the height of the baseball is given by h = 35 − 16 t 2 . To find the initial height, we substitute t = 0 into this equation.
Calculating the Height at t=0 Substituting t = 0 into the equation, we get: h = 35 − 16 ( 0 ) 2 = 35 − 16 ( 0 ) = 35 − 0 = 35 So, the initial height of the baseball is 35 feet.
Choosing the Correct Statement Comparing our result with the given options, we see that the statement 'The height of the baseball is 35 feet at the moment the player begins to leap' is the correct description of the situation.
Examples
Understanding the initial conditions of a projectile's motion is crucial in many real-world scenarios, such as sports and engineering. For example, knowing the initial height of a ball thrown in basketball helps to predict its trajectory and whether it will pass through the hoop. Similarly, in engineering, understanding the initial height of a falling object is essential for designing safety measures and predicting impact forces. By analyzing the initial height and other parameters, we can accurately model and predict the behavior of objects in motion, leading to better decision-making and safer outcomes.
