Question

at the same time, min’s 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?
$-16t^2 + \boxed{} + \boxed{}$
which polynomial represents the difference in the heights of the baseballs $t$ seconds after they are thrown?
$\boxed{}$

Explanation:

First Sub - Question: Polynomial for the height of the ball

Step1: Recall the projectile motion height formula

The general formula for the height \(h(t)\) of an object in vertical motion (under gravity, on Earth, in feet) is \(h(t)=- 16t^{2}+v_{0}t + h_{0}\), where \(v_{0}\) is the initial vertical velocity (in feet per second) and \(h_{0}\) is the initial height (in feet).

Step2: Identify the values of \(v_{0}\) and \(h_{0}\)

Here, the initial vertical velocity \(v_{0} = 20\) feet per second and the initial height \(h_{0}=4\) feet.

Step3: Substitute the values into the formula

Substituting \(v_{0} = 20\) and \(h_{0}=4\) into \(h(t)=-16t^{2}+v_{0}t + h_{0}\), we get \(h(t)=-16t^{2}+20t + 4\). So the first box (the coefficient of \(t\)) is \(20t\) (or just \(20\) in the box as the formula is \(-16t^{2}+\square t+\square\)) and the second box is \(4\).

Second Sub - Question: Polynomial for the difference in heights (assuming Min's ball has some height function, but since we only have the brother's function \(h_{b}(t)=-16t^{2}+20t + 4\), we need to know Min's function. But if we assume Min's ball, for example, has a different initial velocity or height. However, since the problem is about the brother's ball first, and then the difference. But since the first part gives the brother's height as \(-16t^{2}+20t + 4\), if we assume Min's ball, say, has a height function (maybe from a previous part not shown, but let's assume for the sake of the problem that we need to find the difference. But since the first part is to fill \(-16t^{2}+\square+\square\) with \(20t\) and \(4\) (for the brother's ball).

Answer:

(First Sub - Question):
For the polynomial modeling the height of the ball: \(-16t^{2}+20t + 4\), so the first box is \(20t\) (or \(20\) in the coefficient of \(t\) place) and the second box is \(4\).

(Note: For the second sub - question, we need more information about Min's ball's height function. If we assume that Min's ball, for example, has a height function \(h_{m}(t)\) and the brother's is \(h_{b}(t)=-16t^{2}+20t + 4\), the difference would be \(h_{b}(t)-h_{m}(t)\) or \(h_{m}(t)-h_{b}(t)\) depending on the context. Since the problem is incomplete for the second sub - question in terms of Min's ball's function, we can only solve the first part completely here.)