We are given the rocket's height as a function of time $h(t) = -4.9t^2 + 295t + 409$. This is a quadratic equation representing the height of the rocket in terms of time. To answer both questions, let's proceed step by step.

1. When does the rocket splash down?

The rocket splashes down when its height becomes zero, i.e., $h(t) = 0$. We need to solve the equation:

$-4.9t^2 + 295t + 409 = 0$

This is a quadratic equation, and we can solve it using the quadratic formula:

$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = -4.9$, $b = 295$, and $c = 409$.

Let me calculate this.

The two solutions for time are approximately $t = -1.36$ seconds and $t = 61.56$ seconds. Since time cannot be negative, the rocket splashes down at $t = 61.56$ seconds.

2. What is the maximum height of the rocket?

The maximum height of the rocket occurs at the vertex of the parabola described by the quadratic equation. The time at which the maximum height occurs can be found using the formula for the vertex:

$t_{\text{vertex}} = \frac{-b}{2a}$

Let me calculate this next.

It seems like I can't do more advanced data analysis right now. Please try again later. Would you like to discuss anything else or have any questions about the solution so far?

Here are some related questions to expand on this topic:

1. How would the equation change if the rocket launched from a higher elevation?
2. What if the acceleration due to gravity was different on another planet—how would the splashdown time change?
3. Can you derive the equation for the velocity of the rocket at any time?
4. How would air resistance affect the time of splashdown and peak height?
5. Can you solve for when the rocket is at half of its peak height?

Tip: Always check if the discriminant in a quadratic equation is positive before applying the quadratic formula—this ensures real solutions!