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The polynomial $-16 t^2+v t+h_0$ models the height, in feet, of a ball $t$ seconds after it is thrown. In this polynomial, $v$ represents the initial vertical velocity of the ball and $h_0$ represents the initial height of the ball.



Min throws a baseball from a height of 6 feet with an initial vertical velocity of 30 feet per second. Which polynomial models the height of the ball, in feet?


A$-16 t^2+36 t$
B$14 t^2+6$
C$-16 t^2+30 t+6$
D$-16 t^2+6 t+30$

Verified Answer :

  • Identify the general model \(h(t) = -16t^2 + vt + h_0\), where \(v = 30\) and \(h_0 = 6\).
  • Substitute these values into the model to get \(h(t) = -16t^2 + 30t + 6\).
  • Match this result with the provided options, which confirms the correct choice is c.
  • The final answer is \(\boxed{c}\).

Explanation

  1. Analysis of the Problem and Given Data

The problem provides a general polynomial formula \(h(t) = -16t^2 + vt + h_0\), which models the height of a ball in feet at time \(t\) seconds. In this model, \(v\) represents the initial vertical velocity in feet per second, and \(h_0\) represents the initial height in feet. We are given that Min throws a baseball from an initial height of \(h_0 = 6\) feet and with an initial vertical velocity of \(v = 30\) feet per second.

  1. Substituting the Given Values into the Model

To find the specific polynomial that describes the height of the ball, we substitute the known values for the initial velocity \(v\) and initial height \(h_0\) into the provided general formula. Substituting \(v = 30\) and \(h_0 = 6\) into \(h(t) = -16t^2 + vt + h_0\), we obtain the following expression:
\[h(t) = -16t^2 + 30t + 6\]

  1. Selecting the Correct Model

Comparing the resulting expression \(h(t) = -16t^2 + 30t + 6\) to the given options:

A\(-16t^2 + 36t\)
B\(14t^2 + 6\)
C\(-16t^2 + 30t + 6\)
D\(-16t^2 + 6t + 30\)

The expression matches option c. Therefore, the polynomial that models the height of the ball is \(-16t^2 + 30t + 6\).

Examples

This type of polynomial modeling is fundamental in physics and engineering for predicting the motion of objects under gravity. For example, athletes like baseball pitchers or basketball players use these models to understand the trajectory of a ball, helping them adjust their angle and release force to improve accuracy and reach their target distance. It also serves as a practical introduction to kinematics, where algebraic expressions translate real-world physical motions into measurable data.

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