Question

A projectile is thrown upward so that its distance above the ground after t seconds is given by the function h(t) = - 16t^2 + 576t. After how many seconds does the projectile take to reach its maximum height? Show your work for full credit.

Answer 1

h(t) = -16t^2 + 640t This is a "quadratic" (think parabola) and because the coefficient associated with the t^2 term is NEGATIVE, we know it opens downward. Therefore, the vertex is the maximum. . The time (secs) when it reaches the max is: t = -b/(2a) t = -640/(2(-16)) t = -640/(-32) t = 20 seconds

Answer 2

We have to find te vertex of this parabola.
Given a square function:
F(x)=ax²+bx+c
The vertex of this parabola will be the point (-b/2a, f(-b/2a)

In this case:
h(t)=-16t²+576t
a=-16
b=576
-b/2a=-576/-32=18
h(18)=-16(18)²+576(18)=-5184+10368=5184
The vertex will be: (18, 5184)

Because a is negative we have a maximum, if a were positive we have a minimum.
Then we have a minimum at t=18 s; and the maximum heigth would be 5184 m

Answer: The projectile take to reach its maximum heigth in 18 s, and the maximum height would be 5184 m.