Answer:
The maximum height the soccer ball reaches is 9.77 feet.
Step-by-step explanation:
Vertex of a quadratic function:
Suppose we have a quadratic function in the following format:
![f(x) = ax^{2} + bx + c](https://tex.z-dn.net/?f=f%28x%29%20%3D%20ax%5E%7B2%7D%20%2B%20bx%20%2B%20c)
It's vertex is the point ![(x_{v}, y_{v})](https://tex.z-dn.net/?f=%28x_%7Bv%7D%2C%20y_%7Bv%7D%29)
In which
![x_{v} = -\frac{b}{2a}](https://tex.z-dn.net/?f=x_%7Bv%7D%20%3D%20-%5Cfrac%7Bb%7D%7B2a%7D)
![y_{v} = -\frac{\Delta}{4a}](https://tex.z-dn.net/?f=y_%7Bv%7D%20%3D%20-%5Cfrac%7B%5CDelta%7D%7B4a%7D)
Where
![\Delta = b^2-4ac](https://tex.z-dn.net/?f=%5CDelta%20%3D%20b%5E2-4ac)
If a<0, the vertex is a maximum point, that is, the maximum value happens at
, and it's value is
.
Reba kicks a soccer ball off the ground and in the air, with an initial velocity of 25 feet per second.
The formula for the height is:
![h(t) = -16t^2 + v(0)t + h(0)](https://tex.z-dn.net/?f=h%28t%29%20%3D%20-16t%5E2%20%2B%20v%280%29t%20%2B%20h%280%29)
In which v(0) is the initial velocity and s(0) is the initial height.
Initial velocity of 25 feet per second means that ![v(0) = 25](https://tex.z-dn.net/?f=v%280%29%20%3D%2025)
Kicked off the ground means that
. So
![h(t) = -16t^2 + 25t](https://tex.z-dn.net/?f=h%28t%29%20%3D%20-16t%5E2%20%2B%2025t)
Which is a quadratic equation with
.
The maximum value is:
![y_{v} = -\frac{\Delta}{4a}](https://tex.z-dn.net/?f=y_%7Bv%7D%20%3D%20-%5Cfrac%7B%5CDelta%7D%7B4a%7D)
In which
![\Delta = b^2-4ac = 25^2 - 4(-16)(0) = 625](https://tex.z-dn.net/?f=%5CDelta%20%3D%20b%5E2-4ac%20%3D%2025%5E2%20-%204%28-16%29%280%29%20%3D%20625)
![y_{v} = -\frac{\Delta}{4a} = \frac{625}{4(16)} = 9.77](https://tex.z-dn.net/?f=y_%7Bv%7D%20%3D%20-%5Cfrac%7B%5CDelta%7D%7B4a%7D%20%3D%20%5Cfrac%7B625%7D%7B4%2816%29%7D%20%3D%209.77)
The maximum height the soccer ball reaches is 9.77 feet.