To find the time at which the rocket reaches its maximum height, we need to find the vertex of the quadratic equation:

$y = -16x^2 + 247x + 81$

In a quadratic equation of the form $y = ax^2 + bx + c$, the time at which the maximum height occurs is given by the formula for the x-coordinate of the vertex:

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

For your equation, $a = -16$, $b = 247$, and $c = 81$. Let's substitute these values into the formula:

$x = \frac{-247}{2(-16)} = \frac{-247}{-32} = \frac{247}{32}$

Now, calculate the value of $x$:

$x = 7.71875$

Rounding this to the nearest 100th of a second:

$x \approx 7.72 \, \text{seconds}$

So, the rocket will reach its maximum height at 7.72 seconds after launch.

Would you like further explanation on this, or do you have any other questions?

Related Questions:

1. How do you determine the maximum height of the rocket?
2. How does the equation of motion relate to real-world projectile motion?
3. What effect would changing the value of $a$ have on the trajectory?
4. How would you modify the equation if the rocket was launched from a higher or lower point?
5. How does the vertex formula apply to other quadratic functions?

Tip: Remember, the vertex formula is always a helpful tool when analyzing parabolic motion, whether it's projectile motion or any other situation described by a quadratic equation.