Answer:
[tex]h = 227\ ft[/tex]
Step-by-step explanation:
We know that the equation that models the height of a projectile as a function of time is:
[tex]h(t) = -16t ^ 2 +v_0t +h_0[/tex]
Where:
[tex]v_0[/tex] is the initial velocity
[tex]h_0[/tex] is the initial height of the projectile.
In our case, the height of the machine is 2 ft.
Then [tex]h_0 = 2\ ft[/tex]
The initial speed is 120 ft/s.
So the equation of the height for this case is:
[tex]h(t) = -16t ^ 2 + 120t + 2[/tex]
This is a quadratic equation whose main coefficient is negative.
The maximum value of the function is at its vertex.
For a quadratic function of the form:
[tex]at ^ 2 + bt + c[/tex]
the vertex of the equation is given by the expression:
[tex]x =\frac{-b}{2a}[/tex]
[tex]y = f(\frac{-b}{2a})[/tex]
In this case:
[tex]a = -16\\b = 120\\c = 2[/tex]
Then the maximum point occurs instantly:
[tex]t = -\frac{120}{2(-16)}\\\\t = 3.75\ s[/tex]
Finally the maximum atura is:
[tex]h(3.75) = -16(3.75) ^ 2 +120(3.75) + 2[/tex]
[tex]h = 227\ ft[/tex]
