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:
[tex]h=-16t^2+96t+256[/tex]Since we want to know when the ball hits the ground, we will let the height of the ball be h = 0.
[tex]0=-16t^2+96t+256[/tex]
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:
[tex]GCF(-16,96,256)=-16[/tex]We get:
[tex]\begin{gathered} 0=\frac{-16}{-16}t^2+\frac{96}{-16}t+\frac{256}{-16} \\ \\ 0=t^2-6t-16 \end{gathered}[/tex]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:
[tex]0=t^2-8t+2t-16[/tex]Grouping the first two and last two terms gives us:
[tex]0=(t^2-8t)+(2t-16)[/tex]We can factor out a t from the first set, and 2 from the second set, like this:
[tex]0=t(t-8)+2(t-8)[/tex]Finishing the factoring method, we have:
[tex]0=(t-8)(t+2)[/tex]Next, we apply the zero product property to solve for each factor:
[tex]t-8=0\text{ and }t+2=0[/tex]From the first factor, we get:
[tex]t-8=0\rightarrow t=8[/tex]For the second factor:
[tex]t+2=0\rightarrow t=-2[/tex]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."
