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The problem you're working with involves the function \( h(t) = -4.9t^2 + 94t + 199 \), which gives the height of a rocket over time.

### Part 1: Splashdown Time
You correctly identified that the splashdown occurs at \( t = 21.10 \) seconds. This was likely found by solving \( h(t) = 0 \), which represents the height being zero (the point when the rocket hits the ocean).

### Part 2: Maximum Height
It seems there was an error in your calculation for the maximum height. To find the time at which the rocket reaches its peak, you need to calculate the vertex of the parabola. For a quadratic function of the form \( h(t) = at^2 + bt + c \), the time \( t \) at which the peak occurs is given by:

\[
t_{\text{peak}} = \frac{-b}{2a}
\]

For \( h(t) = -4.9t^2 + 94t + 199 \):
- \( a = -4.9 \)
- \( b = 94 \)

Now, calculate the peak time:

\[
t_{\text{peak}} = \frac{-94}{2(-4.9)} = \frac{-94}{-9.8} = 9.59 \, \text{seconds}
\]

At \( t = 9.59 \) seconds, the rocket reaches its maximum height. To find the height at this time, substitute \( t = 9.59 \) into the original equation for \( h(t) \):

\[
h(9.59) = -4.9(9.59)^2 + 94(9.59) + 199
\]
\[
h(9.59) = -4.9(91.94) + 900.46 + 199
\]
\[
h(9.59) = -450.51 + 900.46 + 199 = 648.95 \, \text{meters}
\]

So, the correct maximum height is approximately **648.95 meters**.

Would you like further details or have any questions?

### Further Related Questions:
1. How would you calculate the velocity of the rocket at any given time?
2. What is the acceleration of the rocket at any point in time?
3. How long does it take for the rocket to reach half of its maximum height?
4. What is the significance of the discriminant in solving \( h(t) = 0 \)?
5. How would the equation change if the rocket had a different initial velocity?

#### Tip:
For quadratic functions, the vertex formula is key in finding the maximum or minimum points, which are especially useful in physics problems involving motion.

NASA launches a rocket at t=0 seconds. Its height in meters above sea-level is given by h(t) = -4.9t^2 + 94t + 199. Find the splashdown time and the maximum height reached by the rocket.

Solve the quadratic function h(t) = -4.9t^2 + 94t + 199 to find the rocket's splashdown time and maximum height. Learn the application of parabolas and vertex calculations. Ideal for students in Grades 10-12.

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