In order to determine the truth of these statements, we need to work with the function that describes the height of the ball, $h(t)$. While the exact function for the height of the ball is not provided here, we can reasonably assume it has a general quadratic form like:

$h(t) = -16t^2 + v_0t + h_0$

where:

• $t$ is the time in seconds,
• $v_0$ is the initial velocity of the ball,
• $h_0$ is the initial height of the ball (e.g., the height of the roof),
• The $-16t^2$ term represents the effect of gravity.

Let’s analyze each statement given the context:

1. The height of the ball after 3 seconds is 112 feet.

   To evaluate this, we would need to plug $t = 3$ into the height function. Without knowing the exact function, we cannot confirm this directly. However, if after calculating, $h(3) = 112$, this statement would be true; otherwise, false.

2. The height of the ball after 6 seconds is 224 feet.

   Similarly, we would need to evaluate $h(6)$ to see if it equals 224 feet. If the function gives this value at $t = 6$, then the statement is true. Otherwise, it’s false.

3. The value of $h(3)$ represents the height of the ball after 3 seconds.

   This statement is true by the definition of $h(t)$, which represents the height of the ball at time $t$ seconds. So $h(3)$ represents the height after 3 seconds.

4. The value of $h(6)$ has no meaning as the height of the ball cannot be negative.

   This statement assumes that $h(6)$ results in a negative value. If the height at $t = 6$ seconds is negative, it means the ball has already hit the ground or passed below the initial reference point. Thus, while physically the height cannot be negative, mathematically it still holds meaning in a model like this. The statement is false because negative height values can still indicate that the ball has fallen below the original reference height.

Would you like to explore the exact function and solve it for specific values of $t$? Here are five related questions:

1. What is the exact equation for the height function of the ball?
2. How would the function change if Paul threw the ball from a different height?
3. What is the maximum height the ball reaches?
4. How long does it take for the ball to hit the ground?
5. How would the function look if the ball was thrown downward instead?

Tip: When analyzing projectile motion, quadratic functions describe the relationship between time and height, with gravity causing a downward acceleration.