Answer:
The one that makes more sense for our conclusion is that the rocket crashes approximately 6.299601 seconds after it has been launched given the path of the rocket is
.
Step-by-step explanation:
The rocket has crashed on the ground when the height between the ground and the rocket is 0.
We want to find the time,
, such that the height,
, is 0.
We are going to solve the following equation:
with ![y=0](https://tex.z-dn.net/?f=y%3D0)
![0=-16x^2+100x+5](https://tex.z-dn.net/?f=0%3D-16x%5E2%2B100x%2B5)
Upon comparing this equation to
, I see that I have the following values for
and
:
![a=-16](https://tex.z-dn.net/?f=a%3D-16)
![b=100](https://tex.z-dn.net/?f=b%3D100)
![c=5](https://tex.z-dn.net/?f=c%3D5)
The quadratic formula is:
.
Let's plug in the values we got above now.
![x=\frac{-100 \pm \sqrt{100^2-4(-16)(5)}}{2(-16)}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-100%20%5Cpm%20%5Csqrt%7B100%5E2-4%28-16%29%285%29%7D%7D%7B2%28-16%29%7D)
![x=\frac{-100 \pm \sqrt{10000+320}}{-32}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-100%20%5Cpm%20%5Csqrt%7B10000%2B320%7D%7D%7B-32%7D)
![x=\frac{-100 \pm \sqrt{10320}}{-32}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-100%20%5Cpm%20%5Csqrt%7B10320%7D%7D%7B-32%7D)
![x=\frac{-100 \pm \sqrt{16 \cdot 645}}{-32}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-100%20%5Cpm%20%5Csqrt%7B16%20%5Ccdot%20645%7D%7D%7B-32%7D)
![x=\frac{-100 \pm \sqrt{16} \sqrt{645}}{-32}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-100%20%5Cpm%20%5Csqrt%7B16%7D%20%5Csqrt%7B645%7D%7D%7B-32%7D)
![x=\frac{-100 \pm 4 \sqrt{645}}{-32}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-100%20%5Cpm%204%20%5Csqrt%7B645%7D%7D%7B-32%7D)
![x=\frac{\frac{-100}{4} \pm \frac{4}{4} \sqrt{645}}{\frac{-32}{4}}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B%5Cfrac%7B-100%7D%7B4%7D%20%5Cpm%20%5Cfrac%7B4%7D%7B4%7D%20%5Csqrt%7B645%7D%7D%7B%5Cfrac%7B-32%7D%7B4%7D%7D)
![x=\frac{-25 \pm \sqrt{645}}{-8}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-25%20%5Cpm%20%5Csqrt%7B645%7D%7D%7B-8%7D)
This gives us either:
![x=\frac{-25 + \sqrt{645}}{-8} \text{ or } \frac{-25 - \sqrt{645}}{-8}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-25%20%2B%20%5Csqrt%7B645%7D%7D%7B-8%7D%20%5Ctext%7B%20or%20%7D%20%5Cfrac%7B-25%20-%20%5Csqrt%7B645%7D%7D%7B-8%7D)
Let's punch both of these into the calculator:
![x \approx -0.04961 \text{ or } 6.299601](https://tex.z-dn.net/?f=x%20%5Capprox%20-0.04961%20%5Ctext%7B%20or%20%7D%206.299601)
The one that makes more sense for our conclusion is that the rocket crashes approximately 6.299601 seconds after it has been launched.