To analyze the rocket's motion, we are given the height equation:

$h(t) = -16t^2 + 80t + 576$

Step 1: Find the time to reach maximum height

The time to reach the maximum height occurs at the vertex of the parabola, which is given by:

$t = -\frac{b}{2a}$

Here, $a = -16$, $b = 80$, and $c = 576$. Substituting these values:

$t = -\frac{80}{2(-16)} = \frac{80}{32} = 2.5 \, \text{seconds}.$

So, the rocket reaches its maximum height at $t = 2.5$ seconds.

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Step 2: Find the maximum height

To find the maximum height, substitute $t = 2.5$ into the height equation:

$h(2.5) = -16(2.5)^2 + 80(2.5) + 576$

Calculate step by step:

$h(2.5) = -16(6.25) + 80(2.5) + 576$

$h(2.5) = -100 + 200 + 576 = 676 \, \text{feet}.$

The maximum height is $676 \, \text{feet}$.

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Step 3: Find when the rocket reaches the ground

The rocket reaches the ground when $h(t) = 0$. Solve:

$-16t^2 + 80t + 576 = 0$

Divide the entire equation by $-16$:

$t^2 - 5t - 36 = 0$

Factorize or use the quadratic formula. The quadratic formula is:

$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 1$, $b = -5$, $c = -36$. Substituting:

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

$t = \frac{5 \pm \sqrt{25 + 144}}{2}$

$t = \frac{5 \pm \sqrt{169}}{2}$

$t = \frac{5 \pm 13}{2}$

$t = \frac{18}{2} = 9 \quad \text{or} \quad t = \frac{-8}{2} = -4.$

Since time cannot be negative, $t = 9 \, \text{seconds}$.

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Final Results:

1. Maximum height: $676 \, \text{feet}$.
2. Time to reach maximum height: $2.5 \, \text{seconds}$.
3. Time to hit the ground: $9 \, \text{seconds}$.

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Do you want a detailed explanation of any step or graphing? Here are some related questions:

1. How does changing the initial height affect the results?
2. What happens to the maximum height if the initial velocity is doubled?
3. Can we graph $h(t)$ to visualize the motion of the rocket?
4. How would the results change if the equation included air resistance?
5. How can we calculate the velocity of the rocket at a specific time?

Tip: Always check the discriminant ($b^2 - 4ac$) before solving quadratic equations to confirm real solutions.