The given equation for the height of the rocket is:

$y = -16x^2 + 135x + 65$

This is a quadratic equation in terms of $x$, where $x$ is the time in seconds, and $y$ is the height of the rocket in feet. To find the time at which the rocket reaches its maximum height, we need to determine the vertex of the parabola since the vertex represents the maximum point for a parabola that opens downwards (as 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 (time at which the maximum occurs) is given by the formula:

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

Step 1: Identify the coefficients

From the equation $y = -16x^2 + 135x + 65$:

• $a = -16$
• $b = 135$

Step 2: Apply the vertex formula

$x = \frac{-135}{2(-16)} = \frac{-135}{-32} = 4.21875$

Step 3: Round to the nearest hundredth

The time at which the rocket reaches its maximum height is approximately:

$x = 4.22 \, \text{seconds}$

Would you like further details on this solution or any specific clarification?

Here are 5 related questions:

1. What is the height of the rocket when it reaches its maximum?
2. How would the equation change if the rocket had a different initial speed?
3. Can you find when the rocket returns to the ground (height $y = 0$)?
4. How would the maximum height be affected if gravity were weaker?
5. Can this problem be solved graphically? How would the graph look?

Tip: The vertex of a parabola provides important information about either the minimum or maximum point in physical scenarios involving motion or optimization.