1. ## Polynomial Functions

Hello. I was hoping someone could help me with this problem as it relates to polynomial functions.

A projectile is thrown upward so that its distance above the ground after t seconds is given by the function h(t) = - 16t2 + 576t. After how many seconds does the projectile take to reach its maximum height?

It would be nice if you could give me a full explanation as I truly want to learn how to do this. Thank you so much guys.

2. ## Re: Polynomial Functions

A projectile is thrown upward so that its height above the ground after t seconds is given by the function:
. . $\displaystyle h(t) \:=\: - 16t^2 + 576t$
After how many seconds does the projectile take to reach its maximum height?

I'll assume that you are not familiar with Calculus.

The graph of $\displaystyle h(t)$ is a down-opening parabola.
Its maximum is at its vertex.
The formula for the vertex is: .$\displaystyle t \:=\:\frac{\text{-}b}{2a}$

We have: .$\displaystyle a = -16,\;b = 576$

Hence: .$\displaystyle t \:=\:\frac{\text{-}576}{2(\text{-}16)} \:=\:18$

The projectile takes 18 second to reach maximum height.

[The maximum height is: .$\displaystyle h(18) \:=\:-16(18^2) + 576(18) \:=\: 5184$ units.]

3. ## Re: Polynomial Functions

You have another way on completing this sum using quadratic function
h_(t)=-16t^2+576t
h_((t) )=-16(t^2-36t)
h_((t) )=-16(t^2-36t+324-324)
h_((t) )=-16((t-18)^2-324)
h_((t) )=-16(t-18)^2+5184
(h_((t) ) )_max=(-16(t-18)^2 )_min+5184

(-16(t-18)^2 )_min=0 when t=18
Therefore (h_((t) ) )_max=5184 units