Question

$-16t^{2}+v_{0}t+h_{0}$ models the height, in feet, of a baseball $t$ seconds after it is thrown. in this model, $v_{0}$ represents the initial vertical velocity and $h_{0}$ represents the initial height of the ball.1. a baseball is thrown from a height of 6 feet with an initial vertical velocity of 30 feet per second. what polynomial models the height of the ball, in feet?2. at the same time, alexs little brother throws a baseball from a height of 4 feet with an initial vertical velocity of 20 feet per second. what polynomial models the height of this ball, in feet?3. which polynomial represents the difference in the heights of the baseballs $t$ seconds after they are thrown?

Explanation:

Step1: Identify given height formula

The general height model is $h(t) = -16t^2 + v_0t + h_0$, where $v_0$ = initial vertical velocity, $h_0$ = initial height.

Step2: Write Alex's brother's height model

Substitute $v_0=20$, $h_0=4$:
$h_b(t) = -16t^2 + 20t + 4$

Step3: Write Alex's height model

Substitute $v_0=30$, $h_0=6$:
$h_a(t) = -16t^2 + 30t + 6$

Step4: Calculate height difference

Compute $h_a(t) - h_b(t)$:

$$\begin{align*} (-16t^2 + 30t + 6) - (-16t^2 + 20t + 4) &= -16t^2 + 30t + 6 + 16t^2 - 20t - 4\\ &= ( -16t^2 + 16t^2 ) + (30t - 20t) + (6 - 4) \end{align*}$$

Step5: Simplify the expression

$$\begin{align*} &= 0t^2 + 10t + 2\\ &= 10t + 2 \end{align*}$$

Answer:

1. The polynomial for Alex's brother's ball: $\boldsymbol{-16t^2 + 20t + 4}$ (the blanks are filled with 20 and 4)
2. The polynomial for the height difference: $\boldsymbol{10t + 2}$