Answer:
C: maximum height=62.25 feet; time=3.85 seconds
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
.
Solving a quadratic equation:
Given a second order polynomial expressed by the following equation:
.
This polynomial has roots
such that
, given by the following formulas:
![x_{1} = \frac{-b + \sqrt{\Delta}}{2*a}](https://tex.z-dn.net/?f=x_%7B1%7D%20%3D%20%5Cfrac%7B-b%20%2B%20%5Csqrt%7B%5CDelta%7D%7D%7B2%2Aa%7D)
![x_{2} = \frac{-b - \sqrt{\Delta}}{2*a}](https://tex.z-dn.net/?f=x_%7B2%7D%20%3D%20%5Cfrac%7B-b%20-%20%5Csqrt%7B%5CDelta%7D%7D%7B2%2Aa%7D)
![\Delta = b^{2} - 4ac](https://tex.z-dn.net/?f=%5CDelta%20%3D%20b%5E%7B2%7D%20-%204ac)
In this question:
The height of the ball, after t seconds, is given by the following equation:
![h(t) = -16t^2 + 60t + 6](https://tex.z-dn.net/?f=h%28t%29%20%3D%20-16t%5E2%20%2B%2060t%20%2B%206)
Which is a quadratic equation with ![a = -16, b = 60, c = 6](https://tex.z-dn.net/?f=a%20%3D%20-16%2C%20b%20%3D%2060%2C%20c%20%3D%206)
Maximum height:
Since a < 0, we can find the maximum value of the function. We have that:
![\Delta = 60^{2} - 4(-16)(6) = 3984](https://tex.z-dn.net/?f=%5CDelta%20%3D%2060%5E%7B2%7D%20-%204%28-16%29%286%29%20%3D%203984)
![y_{v} = -\frac{3984}{4(-16)} = 62.25](https://tex.z-dn.net/?f=y_%7Bv%7D%20%3D%20-%5Cfrac%7B3984%7D%7B4%28-16%29%7D%20%3D%2062.25)
The maximum height is of 62.25 feet.
Seconds to reach the ground:
![x_{1} = \frac{-60 + \sqrt{3984}}{2*(-16)} = -0.1](https://tex.z-dn.net/?f=x_%7B1%7D%20%3D%20%5Cfrac%7B-60%20%2B%20%5Csqrt%7B3984%7D%7D%7B2%2A%28-16%29%7D%20%3D%20-0.1)
![x_{2} = \frac{-60 - \sqrt{3984}}{2*(-16)} = 3.85](https://tex.z-dn.net/?f=x_%7B2%7D%20%3D%20%5Cfrac%7B-60%20-%20%5Csqrt%7B3984%7D%7D%7B2%2A%28-16%29%7D%20%3D%203.85)
Since time is a positive measure, 3.85 seconds.
The correct answer is given by option C.