Answer:
The projectile will reach a height of 96 feet after about 0.84 seconds as well as after about 7.16 seconds.
Step-by-step explanation:
The height of a projectile fired upward is given by the formula:
![\displaystyle s = v_{0} t - 16t^2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20s%20%3D%20v_%7B0%7D%20t%20-%2016t%5E2)
Where <em>s</em> is the height in feet, <em>v</em>₀ is the initial velocity, and <em>t</em> is the time in seconds.
Given a projectile with an initial velocity of 128 ft/s, we want to determine how long it will take the projectile to reach a height of 96 feet.
In other words, given that <em>v</em>₀ = 128, find <em>t</em> such that <em>s</em> = 96.
Substitute:
![(96) = (128)t-16t^2](https://tex.z-dn.net/?f=%2896%29%20%3D%20%28128%29t-16t%5E2)
This is a quadratic. First, we can divide both sides by -16:
![-6 = -8t+t^2](https://tex.z-dn.net/?f=-6%20%3D%20-8t%2Bt%5E2)
Isolate the equation:
![t^2 - 8t + 6 = 0](https://tex.z-dn.net/?f=t%5E2%20-%208t%20%2B%206%20%3D%200)
The equation isn't factorable, so we can consider using the quadratic formula:
![\displaystyle t = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20t%20%3D%20%5Cfrac%7B-b%5Cpm%5Csqrt%7Bb%5E2%20-%204ac%7D%7D%7B2a%7D)
In this case, <em>a</em> = 1, <em>b</em> = -8, and <em>c</em> = 6. Substitute:
![\displaystyle t = \frac{-(-8)\pm\sqrt{(-8)^2-4(1)(6)}}{2(1)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20t%20%3D%20%5Cfrac%7B-%28-8%29%5Cpm%5Csqrt%7B%28-8%29%5E2-4%281%29%286%29%7D%7D%7B2%281%29%7D)
Simplify:
![\displaystyle t = \frac{8\pm\sqrt{40}}{2} = \frac{8\pm 2\sqrt{10}}{2} = 4\pm \sqrt{10}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20t%20%3D%20%5Cfrac%7B8%5Cpm%5Csqrt%7B40%7D%7D%7B2%7D%20%3D%20%5Cfrac%7B8%5Cpm%202%5Csqrt%7B10%7D%7D%7B2%7D%20%3D%204%5Cpm%20%5Csqrt%7B10%7D)
Hence, our two solutions are:
![\displaystyle t = 4+\sqrt{10} \approx 7.16\text{ or } t= 4-\sqrt{10} \approx 0.84](https://tex.z-dn.net/?f=%5Cdisplaystyle%20t%20%3D%204%2B%5Csqrt%7B10%7D%20%5Capprox%207.16%5Ctext%7B%20or%20%7D%20t%3D%204-%5Csqrt%7B10%7D%20%5Capprox%200.84)
So, the projectile will reach a height of 96 feet after about 0.84 seconds as well as after about 7.16 seconds.