Let's break down the problem step-by-step.

1. Finding the Splashdown Time

The rocket will splash down when its height $h(t)$ is zero. The height of the rocket as a function of time $t$ is given by: $h(t) = -4.9t^2 + 199t + 101$

To find the splashdown time, we need to solve the equation $h(t) = 0$: $-4.9t^2 + 199t + 101 = 0$

This is a quadratic equation of the form $at^2 + bt + c = 0$. We can use the quadratic formula: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = -4.9$, $b = 199$, and $c = 101$.

Calculating the roots: $t = \frac{-199 \pm \sqrt{199^2 - 4(-4.9)(101)}}{2(-4.9)}$

Let's compute the values to find the splashdown time.

2. Finding the Maximum Height

The maximum height of the rocket is achieved at the vertex of the parabolic function $h(t)$. The time at which the vertex occurs can be found using: $t = \frac{-b}{2a}$

For our equation, $a = -4.9$ and $b = 199$: $t = \frac{-199}{2(-4.9)} = \frac{199}{9.8}$

After finding this time, we can substitute it back into the height function $h(t)$ to find the maximum height.

I'll now calculate both the splashdown time and the peak height.### Results:

1. Splashdown Time: The rocket splashes down at approximately $41.11$ seconds.

2. Maximum Height: The rocket reaches its peak height of approximately $2121.46$ meters above sea level.

Would you like further details or have any questions?

Follow-up Questions:

1. How do you find the time of maximum height using calculus?
2. What is the significance of the discriminant in the quadratic formula?
3. How does the initial velocity affect the maximum height and splashdown time?
4. What would happen if we changed the gravity constant in the equation?
5. Can you graph the function to visually represent the rocket's flight?

Tip: To find the maximum or minimum of a quadratic function, use the vertex formula $t = \frac{-b}{2a}$.