The height $h(t)$ of the rocket is given by the quadratic function:

$h(t) = -4.9t^2 + 46t + 241$

Key Features of the Function

1. Vertex (Maximum Height): The vertex of the parabola represents the maximum height of the rocket. The time $t$ at which this occurs can be found using the formula:

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

   where $a = -4.9$ and $b = 46$.

2. Time of Flight: The rocket will rise until it reaches its maximum height and then fall back down. To find the total time of flight, we need to determine when $h(t) = 0$.

3. Maximum Height: Once we find $t$ for the vertex, we can plug it back into $h(t)$ to find the maximum height.

Steps to Solve

1. Find the vertex time: $t = -\frac{46}{2 \cdot -4.9} = \frac{46}{9.8} \approx 4.69 \text{ seconds}$

2. Calculate maximum height: $h(4.69) = -4.9(4.69)^2 + 46(4.69) + 241$

3. Find the time when the rocket hits the ground: Set $h(t) = 0$ and solve the quadratic equation.

Would you like me to calculate the maximum height and the time of flight in detail?