The function #h(t)=-5t^2+20t+2# gives the approximate height, #h# meters, of a thrown football as a function of the time, #t# seconds, since it was thrown. The ball hit the ground before a receiver could get near it. a) How long was the ball in the air?
How long was the ball in the air?
We are told that
To answer the question, then, we need to know the time when it stopped being in the air.
We are told that the ball hit the ground. So that's what happened when it stopped being airborne. We need to relate that event to the mathematics we're working with.
What can we say about
Now translate this back to the mathematics:
The ball is in the air from
Find the time
That means: solve:
We can solve this by solving:
(Either multiply both sides of the equation by
That's a quadratic equation, so try to factor first.
But don't spend too much time trying to factor, because not every quadratic is easily factorable and that's OK, because we still have the quadratic formula if we need it. We do need it.
We can see that
for the solution
The other solution gives
So the ball was in the air from