Answer:
The maximum height is 784 feet
Step-by-step explanation:
In this problem we use the kinematic equation of the height h of an object as a function of time
[tex]h(t) = -16t ^ 2 + v_0t + h_0[/tex]
Where [tex]v_0[/tex] is the initial velocity and [tex]h_0[/tex] is the initial height.
We know that
[tex]v_0 = 192\ \frac{ft}{sec}[/tex]
[tex]h_0 = 208\ ft.[/tex]
Then the equation of the height is:
[tex]h(t) = -16t ^ 2 + 192t +208[/tex]
For a quadratic function of the form [tex]ax ^ 2 + bx + c[/tex]
where [tex]a <0[/tex]
the maximum height of the function is at its vertex.
The vertice is
[tex]x = -\frac{b}{2a}\\\\y = f(\frac{-b}{2a})[/tex]
In this case
[tex]a = -16\\b = 192\\c = 208[/tex]
Then the vertice is:
[tex]t = -\frac{192}{2(-16)}\\\\t = 6\ sec[/tex]
Now we calculate h (6)
[tex]h(6) = -16(6) ^ 2 +192(6) +208\\\\h(6) = 784\ feet[/tex]
The maximum height is 784 feet
