Question

Givenh= -16t^2+96t+256The ball hits the ground after how many seconds?

Answer 1

We want to find out how long it takes a ball to hit the ground using a quadratic function.

Typically, we can do this by finding the x-intercepts, or zeros of the function.

We are given the function:

`h=-16t^2+96t+256`

Since we want to know when the ball hits the ground, we will let the height of the ball be h = 0.

`0=-16t^2+96t+256`

Any method can be helpful here:

- factoring

- completing the square

- quadratic formula

For this, we can use the factoring method. To begin, let's divide out the greatest common factor:

`GCF(-16,96,256)=-16`

We get:

`\begin{gathered} 0=(-16)/(-16)t^2+(96)/(-16)t+(256)/(-16) \\ \\ 0=t^2-6t-16 \end{gathered}`

Next, we want to find two factors of -16 that add to give us -6.

We would get -8 and 2. We can use those to separate the middle term:

`0=t^2-8t+2t-16`

Grouping the first two and last two terms gives us:

`0=(t^2-8t)+(2t-16)`

We can factor out a t from the first set, and 2 from the second set, like this:

`0=t(t-8)+2(t-8)`

Finishing the factoring method, we have:

`0=(t-8)(t+2)`

Next, we apply the zero product property to solve for each factor:

`t-8=0\text{ and }t+2=0`

From the first factor, we get:

`t-8=0\rightarrow t=8`

For the second factor:

`t+2=0\rightarrow t=-2`

Since this is a real world situation, we need to keep the positive value.

Therefore, we will use t = 8. We interpret this as, "It takes 8 seconds for the ball to hit the ground."