Answer:
Part A)
784 feet in the air (after five seconds).
Part B)
After 12 seconds.
Step-by-step explanation:
The height <em>h</em> (in feet) of a rocket <em>t</em> seconds after being fired is modeled by the function:
![h(t)=-16t^2+160t+384](https://tex.z-dn.net/?f=h%28t%29%3D-16t%5E2%2B160t%2B384)
Part A)
We want to find the rocket's maximum height.
Since our function is a quadratic, the maximum height will occur at its vertex. The vertex of a quadratic is given by:
![\displaystyle \Big(-\frac{b}{2a}\, f\Big(-\frac{b}{2a}\Big)\Big)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5CBig%28-%5Cfrac%7Bb%7D%7B2a%7D%5C%2C%20f%5CBig%28-%5Cfrac%7Bb%7D%7B2a%7D%5CBig%29%5CBig%29)
In this case, a = -16, b = 160, and c = 384.
So, the vertex occurs at:
![\displaystyle t=-\frac{160}{2(-16)}=\frac{160}{32}=5\text{ seconds}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20t%3D-%5Cfrac%7B160%7D%7B2%28-16%29%7D%3D%5Cfrac%7B160%7D%7B32%7D%3D5%5Ctext%7B%20seconds%7D)
The maximum height is reached after five seconds.
Then the maximum height is:
![h(t)_\text{max}=h(5)=-16(5)^2+160(5)+384=784\text{ feet}](https://tex.z-dn.net/?f=h%28t%29_%5Ctext%7Bmax%7D%3Dh%285%29%3D-16%285%29%5E2%2B160%285%29%2B384%3D784%5Ctext%7B%20feet%7D)
Part B)
When the rocket reaches the ground, its height <em>h</em> above the ground will be 0. Hence:
![h(t)_\text{ground}=0=-16t^2+160t+384](https://tex.z-dn.net/?f=h%28t%29_%5Ctext%7Bground%7D%3D0%3D-16t%5E2%2B160t%2B384)
Solve for <em>t</em>. We can first divide both sides by -16:
![t^2-10t-24=0](https://tex.z-dn.net/?f=t%5E2-10t-24%3D0)
Factor:
![(t-12)(t+2)=0](https://tex.z-dn.net/?f=%28t-12%29%28t%2B2%29%3D0)
Zero Product Property:
![t-12=0\text{ or } t+2=0](https://tex.z-dn.net/?f=t-12%3D0%5Ctext%7B%20or%20%7D%20t%2B2%3D0)
Solve for each case:
![t=12\text{ or } t=-2](https://tex.z-dn.net/?f=t%3D12%5Ctext%7B%20or%20%7D%20t%3D-2)
Time cannot be negative. Hence, our only solution is:
![t=12\text{ seconds}](https://tex.z-dn.net/?f=t%3D12%5Ctext%7B%20seconds%7D)
The rocket reaches the ground 12 seconds after it is fired.