We are given the equation for the distance $s$ of the ball from the ground:

$s = 96t - 16t^2$

Part (a) — When will the ball strike the ground?

The ball will strike the ground when the distance $s$ is zero. So, we set the equation for $s$ equal to zero and solve for $t$:

$0 = 96t - 16t^2$

First, factor out $t$:

$0 = t(96 - 16t)$

This gives us two possible solutions:

$t = 0 \quad \text{or} \quad 96 - 16t = 0$

Solving the second equation for $t$:

$96 = 16t$ $t = \frac{96}{16} = 6$

So, the ball will strike the ground at $t = 6$ seconds.

Part (b) — For what time $t$ is the ball more than 80 feet above the ground?

We want to solve for $t$ when the ball is higher than 80 feet. Set the equation for $s$ greater than 80:

$96t - 16t^2 > 80$

First, subtract 80 from both sides:

$96t - 16t^2 - 80 > 0$

Rearranging terms:

$-16t^2 + 96t - 80 > 0$

Divide through by $-16$ to simplify (and reverse the inequality):

$t^2 - 6t + 5 < 0$

Now, solve the quadratic equation $t^2 - 6t + 5 = 0$ using the quadratic formula:

$t = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(5)}}{2(1)}$ $t = \frac{6 \pm \sqrt{36 - 20}}{2}$ $t = \frac{6 \pm \sqrt{16}}{2}$ $t = \frac{6 \pm 4}{2}$

This gives two solutions:

$t = \frac{6 + 4}{2} = 5 \quad \text{and} \quad t = \frac{6 - 4}{2} = 1$

Thus, the ball is more than 80 feet above the ground between $t = 1$ second and $t = 5$ seconds.

Summary:

(a) The ball will strike the ground at 6 seconds.

(b) The ball is more than 80 feet above the ground between 1 second and 5 seconds.

Would you like a more detailed explanation of any part? Here are five related questions:

1. What is the maximum height the ball reaches?
2. How would the problem change if the initial velocity were different?
3. Can you calculate the speed of the ball when it hits the ground?
4. What is the velocity of the ball at $t = 2$ seconds?
5. How would this scenario differ if we included air resistance?

Tip: When solving inequalities involving quadratic equations, factorizing or using the quadratic formula often helps in finding the boundary points.