To solve this problem, we need to analyze the given height function for the ball, which is:

$h(t) = -16t^2 + 24t + 8$

1. Finding the maximum height

The height function is a quadratic function of the form $h(t) = at^2 + bt + c$, where:

• $a = -16$
• $b = 24$
• $c = 8$

For any quadratic function $ax^2 + bx + c$, the maximum or minimum occurs at $t = \frac{-b}{2a}$.

Here, since the coefficient $a = -16$ is negative, the parabola opens downward, and thus the vertex represents the maximum height.

So, the time $t_{\text{max}}$ when the ball reaches the maximum height is: $t_{\text{max}} = \frac{-b}{2a} = \frac{-24}{2(-16)} = \frac{24}{32} = 0.75 \text{ seconds}$

Now, substitute $t = 0.75$ into the height function to find the maximum height $h_{\text{max}}$: $h(0.75) = -16(0.75)^2 + 24(0.75) + 8$ $h(0.75) = -16(0.5625) + 18 + 8$ $h(0.75) = -9 + 18 + 8 = 17 \text{ feet}$

Conclusion for Part 1:

The maximum height the ball reaches is 17 feet.

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Would you like to go over the remaining parts of the problem about how long it takes for the ball to descend to 4 feet? Or do you have any further questions on this part? Let me know!

Here are some related questions:

1. What is the formula to calculate the time for the maximum or minimum point of a quadratic function?
2. How do we determine the direction of a parabola (upward or downward)?
3. How would you calculate the time it takes for an object to reach a certain height given a quadratic equation?
4. What happens to the vertex of the quadratic function when the value of $a$ changes?
5. How do we graph a quadratic function to visually determine the maximum or minimum?

Tip: When solving problems with quadratic functions, always check if the parabola opens upwards or downwards by looking at the sign of $a$. This will tell you whether you're finding a minimum or maximum!