The height of the rocket is modeled by the quadratic equation:

$y = -16x^2 + 240x + 96$

To find the maximum height, we need to find the vertex of the parabola, as the vertex represents the maximum point for a downward-opening parabola (since the coefficient of $x^2$ is negative).

For a quadratic equation in the form $y = ax^2 + bx + c$, the x-coordinate of the vertex is given by:

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

In our equation:

• $a = -16$
• $b = 240$

Let's calculate the time $x$ at which the maximum height is reached:

$x = -\frac{240}{2 \times (-16)} = \frac{240}{32} = 7.5 \text{ seconds}$

Now, substitute $x = 7.5$ back into the original equation to find the maximum height $y$:

$y = -16(7.5)^2 + 240(7.5) + 96$

Let me compute that.The maximum height reached by the rocket is 996.0 feet.

Would you like further details or have any questions?

Here are five related questions:

1. How would air resistance affect the rocket's height?
2. Can you find the time it takes for the rocket to hit the ground after reaching its maximum height?
3. How would the maximum height change if the initial velocity was increased?
4. What is the significance of the $x$-coordinate of the vertex in a quadratic function?
5. How can you calculate the total duration of the rocket's flight using this equation?

Tip: The vertex formula $-\frac{b}{2a}$ is essential in finding the maximum or minimum value of any quadratic equation.