The problem shown in the image involves finding the time it takes for a ball to hit the ground when dropped from a height of $576$ feet. The equation given is:

$h = -16t^2 + 576$

where $h$ is the height in feet and $t$ is the time in seconds. To find the time the ball hits the ground, set $h = 0$ and solve for $t$:

Step-by-Step Solution:

1. Set $h = 0$: $0 = -16t^2 + 576$

2. Simplify the equation: $16t^2 = 576$

3. Divide by 16: $t^2 = 36$

4. Solve for $t$: $t = \sqrt{36} \quad \text{or} \quad t = -\sqrt{36}$ Since time cannot be negative, $t = 6$.

Final Answer:

The ball takes 6 seconds to hit the ground.

Would you like a breakdown of solving quadratic equations or have additional questions?

Here are 5 related questions to deepen your understanding:

1. What are the steps for solving a quadratic equation in general?
2. How does gravity influence the quadratic term in the height equation?
3. If the ball were thrown upwards, how would the equation change?
4. Can you solve for the maximum height reached by an object in free fall?
5. What would happen if the height were measured in meters instead of feet?

Tip: Always check the units in physics problems—mixing units can lead to incorrect results.