The given equation for the height of the rocket is:

$y = -16x^2 + 186x + 75$

This is a quadratic equation in the form $y = ax^2 + bx + c$, where $a = -16$, $b = 186$, and $c = 75$. The graph of this equation is a parabola that opens downward since $a < 0$. To find the maximum height of the rocket, we need to determine the vertex of the parabola, which represents the maximum point.

The formula to find the $x$-coordinate of the vertex is:

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

Substituting the values of $a$ and $b$:

$x = \frac{-186}{2(-16)} = \frac{-186}{-32} = 5.8125$

Now that we have the time $x = 5.8125$ seconds at which the rocket reaches its maximum height, we substitute this value back into the original equation to find the maximum height $y$:

$y = -16(5.8125)^2 + 186(5.8125) + 75$

Let's calculate this.The rocket reaches its maximum height at approximately $5.8$ seconds, and the maximum height is approximately $615.6$ feet.

Would you like further details or explanations on any part of this solution?

Here are 5 related questions:

1. How does the value of $a$ affect the shape of the parabola?
2. What would happen to the height equation if gravity were different?
3. How would we find the time when the rocket returns to the ground?
4. Can we determine the initial velocity of the rocket from the given equation?
5. How does the height change after the rocket reaches its maximum?

Tip: The vertex of a parabola provides key information, such as the maximum or minimum value of the function.