To determine the time at which the object reaches its maximum height, we need to analyze the given quadratic function [tex]\( h(t) = -16t^2 + 96t + 6 \)[/tex]. Quadratic functions are generally described by the equation [tex]\( h(t) = at^2 + bt + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants. In this case:
- [tex]\( a = -16 \)[/tex]
- [tex]\( b = 96 \)[/tex]
- [tex]\( c = 6 \)[/tex]
For any quadratic function of the form [tex]\( h(t) = at^2 + bt + c \)[/tex], the vertex of the parabola will give us the maximum or minimum point. Since the coefficient [tex]\( a \)[/tex] is negative, our parabola opens downwards, indicating that the vertex provides the maximum height.
The formula to find the time [tex]\( t \)[/tex] at which the maximum height is reached for a quadratic function [tex]\( at^2 + bt + c \)[/tex] is given by:
[tex]\[ t = \frac{-b}{2a} \][/tex]
Plugging in the coefficients [tex]\( a = -16 \)[/tex] and [tex]\( b = 96 \)[/tex]:
[tex]\[ t = \frac{-96}{2 \times -16} \][/tex]
[tex]\[ t = \frac{-96}{-32} \][/tex]
[tex]\[ t = 3 \][/tex]
So, the object reaches its maximum height after 3 seconds.
The correct answer is:
- 3 seconds
