The equation below models the height in feet, $h$, of a softball $t$ seconds after it is hit by a batter.
$h=-16 t^2+105 t+3$
Why is the height of the ball a function of time?
A. Each value of time is associated with exactly one height.
B. Each height of the ball is associated with exactly one moment in time.
C. There is exactly one time at which the maximum height is attained.
D. There is a maximum and minimum height that the ball can reach.
Answer (1)
A function requires each input to have only one output.
In this problem, time is the input and height is the output.
Each value of time corresponds to exactly one height.
Therefore, the height of the ball is a function of time because each time value is associated with only one height value. $\boxed{Each value of time is associated with exactly one height.}
Explanation
Understanding the Problem The problem states that the height h of a softball t seconds after it is hit is modeled by the equation h = − 16 t 2 + 105 t + 3 . We need to determine why the height of the ball is a function of time.
Understanding the Definition of a Function Recall that a function is a relation where each input has only one output. In this case, the input is time ( t ) and the output is height ( h ). We need to choose the answer that correctly reflects this definition.
Analyzing the Options Let's analyze the given options:
Each value of time is associated with exactly one height: This statement aligns with the definition of a function. For every given time, there is only one corresponding height.
Each height of the ball is associated with exactly one moment in time: This is not necessarily true because the ball can be at the same height at two different times (once on the way up and once on the way down).
There is exactly one time at which the maximum height is attained: While true for this specific quadratic function, this doesn't explain why the height is a function of time in general. It's a property of the function, not the definition of a function.
There is a maximum and minimum height that the ball can reach: This is also a property of the function, not the definition of a function.
Determining the Correct Answer The correct answer is that each value of time is associated with exactly one height. This is because for any given time t , the equation h = − 16 t 2 + 105 t + 3 will produce only one value for the height h .
Examples
Imagine you're tracking the altitude of a drone over time. The drone's height at any given second is a function of time because at each specific second, the drone can only be at one particular height. This concept is crucial in various fields like aviation, robotics, and even economics, where understanding how one variable (like time) uniquely determines another (like height or price) is essential for prediction and control.
