Answer
The rock will reach a maximum height of $6.25$ feet above the ground $0.625$ of a second after its release.
Work Step by Step
The function has degree $2$ so it is quadratic and its graph is a parabola.
Since the coefficient of the squared term is negative, then the parabola opens downward.
A parabola that opens downward has its vertex as its highest point.
The maximum height of the rock will be the y-coordinate of the vertex of the parabola, while the time it will take the rock to reach its highest point will be the x-coordinate of the vertex.
RECALL:
The vertex of a parabola is at $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$.
The function has $a= -16$ and $b= 20$.
Thus, the x-coordinate of the function's vertex is at:
$x=-\frac{b}{2a}
\\x=-\frac{20}{2(-16)}
\\x=-\frac{20}{-32}
\\x=\frac{5}{8}=0.625$
Thus, the rock will reach its maximum height $0.625$ seconds after its release.
Find the y-coordinate of the vertex by evaluating $f(0.625)$:
$f(t)=-16t^2+20t
\\f(0.625)=-16(0.625)^2+20(0.625)
\\f(0.625)=6.25$
Thus, the rock will reach a maximum height of $6.25$ feet.
