To solve the two questions:

The height of the rocket as a function of time is given by:

$h(t) = -4.9t^2 + 205t + 304$

1. Time of Splashdown:

The rocket splashes down when its height becomes zero ($h(t) = 0$). Solve for $t$ using the quadratic equation:

$-4.9t^2 + 205t + 304 = 0$

The quadratic formula is:

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

Here:

• $a = -4.9$
• $b = 205$
• $c = 304$

Substitute into the formula:

$t = \frac{-205 \pm \sqrt{205^2 - 4(-4.9)(304)}}{2(-4.9)}$

$t = \frac{-205 \pm \sqrt{42025 + 5958.4}}{-9.8}$

$t = \frac{-205 \pm \sqrt{47983.4}}{-9.8}$

$t = \frac{-205 \pm 219.02}{-9.8}$

This gives two solutions: $t = \frac{-205 + 219.02}{-9.8} \quad \text{and} \quad t = \frac{-205 - 219.02}{-9.8}$

Calculating each: $t_1 = \frac{14.02}{-9.8} \approx -1.43 \, \text{(not valid as time cannot be negative)}$

$t_2 = \frac{-424.02}{-9.8} \approx 43.27 \, \text{seconds}$

So, the rocket splashes down after approximately 43.27 seconds.

2. Peak Height:

The rocket reaches its peak when the vertical velocity becomes zero. The velocity is the derivative of $h(t)$:

$v(t) = \frac{dh}{dt} = -9.8t + 205$

Set $v(t) = 0$ to find the time at the peak: $-9.8t + 205 = 0$

$t = \frac{205}{9.8} \approx 20.92 \, \text{seconds}$

Substitute $t = 20.92$ into $h(t)$ to find the peak height:

$h(20.92) = -4.9(20.92)^2 + 205(20.92) + 304$

First, calculate $(20.92)^2 \approx 437.86$: $h(20.92) = -4.9(437.86) + 205(20.92) + 304$

$h(20.92) = -2145.51 + 4288.6 + 304$

$h(20.92) \approx 2447.09 \, \text{meters}$

So, the rocket peaks at approximately 2447.09 meters.

Final Answers:

1. The rocket splashes down after 43.27 seconds.
2. The rocket peaks at 2447.09 meters above sea-level.

Would you like detailed step-by-step explanations for these solutions?

Related Questions:

1. What is the initial height of the rocket at $t = 0$?
2. How long does it take for the rocket to reach 1000 meters?
3. What is the velocity of the rocket at splashdown?
4. How would the results change if air resistance is considered?
5. What is the acceleration of the rocket throughout its motion?

Tip:

When solving quadratic equations, always check if the discriminant ($b^2 - 4ac$) is positive to ensure real solutions.