Answer:
a) 64 feet
b) 3 seconds
Step-by-step explanation:
a)
The maximum height of
can be bound by finding the y-coordinate of the vertex of
.
Compare this equation to
to find the values of
.
![a=-16](https://tex.z-dn.net/?f=a%3D-16)
![b=32](https://tex.z-dn.net/?f=b%3D32)
.
The x-coordinate of the vertex can be found by evaluating:
![\frac{-b}{2a}=\frac{-32}{2(-16)](https://tex.z-dn.net/?f=%5Cfrac%7B-b%7D%7B2a%7D%3D%5Cfrac%7B-32%7D%7B2%28-16%29)
![\frac{-b}{2a}=\frac{-32}{-32}](https://tex.z-dn.net/?f=%5Cfrac%7B-b%7D%7B2a%7D%3D%5Cfrac%7B-32%7D%7B-32%7D)
![\frac{-b}{2a}=1](https://tex.z-dn.net/?f=%5Cfrac%7B-b%7D%7B2a%7D%3D1)
So the x-coordinate of the vertex is 1.
The y-coordinate can be found be evaluating
at
:
![y=-16(1)^2+32(1)+48](https://tex.z-dn.net/?f=y%3D-16%281%29%5E2%2B32%281%29%2B48)
![y=-16+32+48](https://tex.z-dn.net/?f=y%3D-16%2B32%2B48)
![y=16+48](https://tex.z-dn.net/?f=y%3D16%2B48)
![y=64](https://tex.z-dn.net/?f=y%3D64)
So the maximum height of the rocket is 64 ft high.
b)
When the rocket hit's the ground the height that the rocket will be from the ground is 0 ft.
So we are trying to find the second t such that:
![0=-16t^2+32t+48](https://tex.z-dn.net/?f=0%3D-16t%5E2%2B32t%2B48)
I'm going to divide both sides by -16:
![0=t^2-2t-3](https://tex.z-dn.net/?f=0%3Dt%5E2-2t-3)
Now we need to find two numbers that multiply to be -3 and add to be -2.
Those numbers are -3 and 1 since (-3)(1)=-3 and (-3)+(1)=-2.
![0=(t-3)(t+1)](https://tex.z-dn.net/?f=0%3D%28t-3%29%28t%2B1%29)
This implies we have either
or ![t+1=0](https://tex.z-dn.net/?f=t%2B1%3D0)
The first equation can be solved by adding 3 on both sides:
.
The second equation can be solved by subtracting 1 on both sides:
.
So when
seconds, is when the rocket has hit the ground.