The equation representing the height $y$ of the rocket as a function of time $x$ in seconds is:

$y = -16x^2 + 121x + 135$

This is a quadratic equation in the form $y = ax^2 + bx + c$, where $a = -16$, $b = 121$, and $c = 135$.

To find the maximum height, we need to determine the vertex of this parabola. The $x$-coordinate of the vertex (which gives the time when the rocket reaches its maximum height) can be found using the formula:

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

Step 1: Find the time $x$ when the maximum height occurs.

$x = \frac{-121}{2(-16)} = \frac{-121}{-32} = 3.78125 \, \text{seconds}$

Step 2: Find the maximum height by substituting $x = 3.78125$ into the original equation.

Now, calculate the height at this time:

$y = -16(3.78125)^2 + 121(3.78125) + 135$

Let me calculate that for you.The maximum height reached by the rocket is approximately 363.8 feet, to the nearest tenth.

Would you like more details or have any other questions?

Here are 5 related questions you might find interesting:

1. What is the height of the rocket after 5 seconds?
2. How long does it take for the rocket to hit the ground after launch?
3. At what time does the rocket reach a height of 200 feet?
4. What is the initial velocity of the rocket based on this equation?
5. How would air resistance affect the equation for the rocket's height?

Tip: The vertex formula $x = \frac{-b}{2a}$ is useful for finding the maximum or minimum points in any quadratic equation.