Question

at the same time, mins 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? -16t²+<u> </u>t+<u> </u> which polynomial represents the difference in the heights of the baseballs t seconds after they are thrown? <u> </u>

Explanation:

Step1: Recall the height - time formula for vertical motion

The general formula for the height $h(t)$ of an object in vertical - motion under the influence of gravity is $h(t)=- 16t^{2}+v_{0}t + h_{0}$, where $-16$ is half of the acceleration due to gravity in feet per second squared ($a=-32$ ft/s², so $\frac{a}{2}=-16$), $v_{0}$ is the initial vertical velocity, and $h_{0}$ is the initial height.

Step2: Identify the values of $v_{0}$ and $h_{0}$

Given that the initial height $h_{0} = 4$ feet and the initial vertical velocity $v_{0}=20$ feet per second. Substitute these values into the formula:
$h(t)=-16t^{2}+20t + 4$.

Step3: Assume another ball's height function (not given fully in the problem - assume we need to find the difference)

Let's assume we want to find the difference between two height functions. But first, we have the height function of the first ball $h_{1}(t)=-16t^{2}+20t + 4$. If we assume another ball's height function is $h_{2}(t)=-16t^{2}+36t$ (from the given expressions), then the difference $\Delta h(t)=h_{1}(t)-h_{2}(t)$.
\[

$$\begin{align*} \Delta h(t)&=(-16t^{2}+20t + 4)-(-16t^{2}+36t)\\ &=-16t^{2}+20t + 4 + 16t^{2}-36t\\ &=( - 16t^{2}+16t^{2})+(20t-36t)+4\\ &=-16t + 4 \end{align*}$$

\]

Answer:

The polynomial that models the height of the ball thrown by Min's little brother is $-16t^{2}+20t + 4$. The polynomial that represents the difference in the heights of the baseballs is $-16t + 4$.