To determine the time at which a ball thrown vertically upward reaches its maximum height, we need to analyze the height function given:
[tex]\[ h(t) = -16t^2 + 32t \][/tex]
This function represents a parabola opening downward, as indicated by the negative coefficient of the [tex]\( t^2 \)[/tex] term. The maximum height of a parabola that opens downward occurs at its vertex.
For a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], the vertex occurs at [tex]\( t = -\frac{b}{2a} \)[/tex]. Here, the coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex] can be identified from the given function:
[tex]\[ a = -16 \][/tex]
[tex]\[ b = 32 \][/tex]
Now, we use the vertex formula to find the time at which the maximum height is reached:
[tex]\[ t = -\frac{b}{2a} \][/tex]
[tex]\[ t = -\frac{32}{2 \times (-16)} \][/tex]
[tex]\[ t = -\frac{32}{-32} \][/tex]
[tex]\[ t = 1 \][/tex]
Therefore, it will take the ball 1 second to reach its maximum height.
So, the correct answer is:
- 1 second
