To solve this problem, we need to find the maximum height of the rocket using the quadratic equation provided:

$y = -16x^2 + 198x + 77$

where:

• $y$ is the height of the rocket in feet,
• $x$ is the time in seconds after the rocket is launched.

Step 1: Identify the vertex of the quadratic equation

Since this is a parabola that opens downward (the coefficient of $x^2$ is negative), the maximum height is located at the vertex of the parabola. The formula to find the time $x$ at which the vertex occurs is:

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

where $a = -16$ and $b = 198$.

Step 2: Calculate the time at maximum height

$x = -\frac{198}{2 \times (-16)} = \frac{198}{32} = 6.1875$

So, the rocket reaches its maximum height at $x = 6.1875$ seconds.

Step 3: Calculate the maximum height

Substitute $x = 6.1875$ into the equation for $y$:

$y = -16(6.1875)^2 + 198(6.1875) + 77$

Now, I'll calculate this step to find the maximum height.

The rocket reaches its maximum height at $x = 6.2$ seconds (rounded to the nearest tenth).

The maximum height is $689.6$ feet (rounded to the nearest tenth of a foot).

Would you like more details or have any questions? Here are five related questions you might find helpful:

1. How do you find the time of flight for the rocket using this equation?
2. How would the equation change if the rocket were launched from a different height?
3. What if the coefficient of $x^2$ was positive—how would the graph and height change?
4. How can you determine when the rocket hits the ground again?
5. How do you interpret the constants in the quadratic equation for real-world applications?

Tip: When solving quadratic equations, always remember that the vertex can represent the maximum or minimum point, depending on whether the parabola opens upward or downward.