Answer:
Part a) The time it take to the rocket to hit its maximum height is 4 seconds
Part b) The maximum height is 256 feet
Part c) The rocket take 8 seconds to hit the ground
Part d) x=4
Step-by-step explanation:
we have the function
[tex]h(t)=-16t^2+128t[/tex]
This is a vertical parabola open downward
The vertex is a maximum
Part a) we know that
The time it take to the rocket to hit its maximum height is equal to the x-coordinate of the vertex
Find the vertex
Convert the quadratic equation into vertex form
[tex]h(t)=-16t^2+128t[/tex]
Factor the leading coefficient
[tex]h(t)=-16(t^2-8t)[/tex]
Complete the square
[tex]h(t)=-16(t^2-8t+16)+256[/tex]
Rewrite as perfect square
[tex]h(t)=-16(t-4)^2+256[/tex]
The vertex is the point (4,256)
therefore
The time it take to the rocket to hit its maximum height is 4 seconds
Part b) we know that
The maximum height is equal to the y-coordinate of the vertex
therefore
The maximum height is 256 feet
Part c) we know that
The rocket hit the ground when the height is equal to zero
so
For h=0
[tex]0=-16(t-4)^2+256[/tex]
[tex]16(t-4)^2=256[/tex]
[tex](t-4)^2=16[/tex]
[tex](t-4)=\pm4[/tex]
[tex]t=4\pm4[/tex]
so
t=0, t=8
therefore
The rocket take 8 seconds to hit the ground
Part d) we know that
the equation of the axis of symmetry is equal to the x-coordinate of the vertex
therefore
x=4
