To find the maximum height of the rocket, we use the given quadratic equation:

$y = -16x^2 + 124x + 110$

This is a parabola opening downwards because the coefficient of $x^2$ is negative ($-16$). The maximum height is achieved at the vertex of the parabola.

Step 1: Formula for the vertex

The time at which the maximum height occurs is given by:

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

Here, $a = -16$, $b = 124$, and $c = 110$.

$x = -\frac{124}{2(-16)} = \frac{124}{32} = 3.875 \, \text{seconds}.$

Step 2: Calculate maximum height

Substitute $x = 3.875$ into the equation to find the maximum height $y$:

$y = -16(3.875)^2 + 124(3.875) + 110$

First, calculate each term: $(3.875)^2 = 15.015625, \quad -16(15.015625) = -240.25, \quad 124(3.875) = 480.5$

$y = -240.25 + 480.5 + 110 = 350.25 \, \text{feet}.$

Final Answer:

The maximum height of the rocket is 350.3 feet (to the nearest tenth).

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Would you like a detailed explanation of any part of the process?

Related Questions:

1. How do you find the vertex of a quadratic equation in general?
2. What is the significance of the sign of the $a$ term in the quadratic equation?
3. Can you calculate the time the rocket takes to hit the ground using this equation?
4. How does the initial velocity affect the trajectory of the rocket?
5. What is the physical interpretation of the constant term 110 in the equation?

Tip: The vertex formula is a crucial tool for finding the maximum or minimum value of a quadratic function in applications like projectile motion.