1. ## Ball Above Ground

A ball is thrown vertically upward with an initial velocity of 80 feet per second. The distance s (in feet) of the ball from the ground after t seconds is s = 80t - 16t^2. For what time interval is the ball less than 64 feet above the ground?

2. Originally Posted by symmetry
A ball is thrown vertically upward with an initial velocity of 80 feet per second. The distance s (in feet) of the ball from the ground after t seconds is s = 80t - 16t^2. For what time interval is the ball less than 64 feet above the ground?
If returns back down at,
$80t-16t^2=0$
$16t(5-t)=0$
$t=0,5$

We need for $0\leq t\leq 5$
To satisfy,
$80t-16t^2 \leq 64$
Thus,
$16t^2-80t + 64 \geq 0$
$t^2 - 5t + 4 \geq 0$
$(t-1)(t-4) \geq 0$
We have two cases,
1)Both positive.
$t-1>0$
$t-4>0$
Thus,
$t>4$
2)Both negative,
$t-1<0$
$t-4<0$
Thus,
$t<1$.

Thus, the time intervals are:
$0\leq t\leq 1 \mbox{ and } 4\leq t \leq 5$

3. Hello, symmetry!

I'll baby-step through the explanation . . .

A ball is thrown vertically upward with an initial velocity of 80 ft/sec.
The height $s$ (in feet) of the ball after $t$ seconds is: $s \:= \:80t - 16t^2$
For what time interval is the ball less than 64 feet above the ground?

When is the height of the ball exactly 64 feet?

We have: . $80t - 16t^2 \:=\:60\quad\Rightarrow\quad t^2 - 5t + 4 \:=\:0$
. . $(t - 1)(t - 4)\:=\:0\quad\Rightarrow\quad t \:=\:1,\,4$

So the ball is takes 1 second to reach a height of 64 feet.
. . The time interval is: . $\boxed{[0,\,1)}$

Then it goes beyond 64 feet to its maximum height, then falls back down,
. . passing the 64-foot mark at $t = 4$.
So it is again below 64 feet from $t = 4$ until it hits the ground.
. . Uh-oh! .When does it hit the ground?

The ball is on the ground when $s = 0$.
So we have: . $80t - 16t^2\:=\:0$
. . Then: . $16t(5 - t)\:=\:0\quad\Rightarrow\quad t \:=\:0,\,5$

So the ball is on the ground when $t = 0$ (of course!)
. . and again when $t = 5$.

So it is below 64 feet for last second: . $\boxed{(4,\,5]}$

4. ## ok

I thank you BOTH for your replies and guidance through the question.