We are given the height function of the rocket as:

$h(t) = -4.9t^2 + 277t + 395$

1. Time of Splashdown:

To find the time when the rocket splashes down, we set $h(t) = 0$ (since it will be at sea level when it splashes down).

$-4.9t^2 + 277t + 395 = 0$

We solve this quadratic equation using the quadratic formula:

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

Where:

• $a = -4.9$
• $b = 277$
• $c = 395$

Substitute the values into the quadratic formula:

$t = \frac{-277 \pm \sqrt{277^2 - 4(-4.9)(395)}}{2(-4.9)}$ $t = \frac{-277 \pm \sqrt{76729 - (-7746)}}{-9.8}$ $t = \frac{-277 \pm \sqrt{84475}}{-9.8}$ $t = \frac{-277 \pm 290.66}{-9.8}$

Now, solve for the two possible values of $t$:

1. $t_1 = \frac{-277 + 290.66}{-9.8} = \frac{13.66}{-9.8} \approx -1.396$ (not physically meaningful as time cannot be negative).
2. $t_2 = \frac{-277 - 290.66}{-9.8} = \frac{-567.66}{-9.8} \approx 57.96$

Thus, the rocket splashes down at 57.96 seconds.

2. Maximum Height (Peak):

The maximum height occurs at the vertex of the parabola, which is given by the formula for the time at the vertex of a parabola:

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

Substitute the values of $a = -4.9$ and $b = 277$:

$t_{\text{vertex}} = \frac{-277}{2(-4.9)} = \frac{-277}{-9.8} \approx 28.3 \text{ seconds}$

Now, substitute $t = 28.3$ into the height equation to find the maximum height:

$h(28.3) = -4.9(28.3)^2 + 277(28.3) + 395$ $h(28.3) = -4.9(801.29) + 277(28.3) + 395$ $h(28.3) = -3936.34 + 7830.11 + 395$ $h(28.3) \approx 4288.77 \text{ meters}$

Thus, the rocket reaches a maximum height of 4288.77 meters above sea level.

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Final Answers:

• The rocket splashes down after 57.96 seconds.
• The rocket reaches a peak height of 4288.77 meters.

Would you like more details or have any questions on this?

Further Questions:

1. How does the coefficient $-4.9$ affect the rocket's trajectory?
2. How do you calculate the maximum height of a parabola?
3. What is the significance of the negative time solution in the quadratic equation?
4. How would you graph this quadratic function to visualize the rocket's path?
5. What would happen if the value of $b$ were increased?

Tip: When solving quadratic equations, always check both possible solutions, but discard those that don't make sense in the context (e.g., negative time in this case).