NASA launches a rocket at  
t
=
0
  seconds. Its height, in meters above sea-level, as a function of time is given by  
h
(
t
)
=
−
4.9
t
2
+
295
t
+
409
 .

Assuming that the rocket will splash down into the ocean, at what time does splashdown occur?

The rocket splashes down after 
   seconds.

How high above sea-level does the rocket get at its peak?

The rocket peaks at 
   meters above sea-level.

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!

NASA launches a rocket at t = 0 seconds. Its height, in meters above sea-level, as a function of time is given by h(t) = -4.9t^2 + 295t + 409. Assuming that the rocket will splash down into the ocean, at what time does splashdown occur? How high above sea-level does the rocket get at its peak?

Learn how to solve the quadratic equation for a NASA rocket's height function, h(t) = -4.9t^2 + 295t + 409, to determine when the rocket splashes down into the ocean and its maximum height. This problem covers key algebraic concepts such as the quadratic formula and vertex calculations. Suitable for students in Grades 9-12.

Math Problem Statement

NASA launches a rocket at
t

0 seconds. Its height, in meters above sea-level, as a function of time is given by
h ( t )

Solution

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