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)=-4.9t2+208t+346.

To analyze the height function of the rocket, given by \( h(t) = -4.9t^2 + 208t + 346 \), we can explore various aspects:

1. **Maximum Height**: To find the time at which the rocket reaches its maximum height, we can use the vertex formula for a quadratic function, \( t = -\frac{b}{2a} \), where \( a = -4.9 \) and \( b = 208 \).

2. **Height at Specific Times**: We can calculate the height at specific time intervals, such as at \( t = 0 \) seconds (launch), \( t = 10 \) seconds, etc.

3. **Time of Flight**: To determine when the rocket hits the ground, we can set \( h(t) = 0 \) and solve for \( t \).

4. **Graph of the Function**: Plotting the function will give us a visual representation of the rocket's flight.

5. **Acceleration**: The acceleration due to gravity is constant in this function, which is indicated by the coefficient of \( t^2 \).

### Steps to Find Key Values

1. **Maximum Height Calculation**:
   \[
   t = -\frac{b}{2a} = -\frac{208}{2 \cdot -4.9} \approx 21.2 \text{ seconds}
   \]
   Substituting \( t \) back into the height function will give the maximum height.

2. **Height at Launch (\( t = 0 \))**:
   \[
   h(0) = 346 \text{ meters}
   \]

3. **Ground Impact Time Calculation**:
   Set \( h(t) = 0 \):
   \[
   -4.9t^2 + 208t + 346 = 0
   \]
   Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Would you like to calculate any specific values or delve deeper into a particular aspect? Here are some questions to consider:

1. What is the maximum height of the rocket?
2. At what time does the rocket hit the ground?
3. Can we graph the function for a visual representation?
4. How does the acceleration affect the rocket's motion?
5. What other factors might influence the rocket's flight path?

**Tip**: Understanding the properties of quadratic functions can greatly help in analyzing projectile motion!

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)=-4.9t2+208t+346.

Explore the height function of a rocket launched by NASA, represented by h(t) = -4.9t² + 208t + 346. This problem examines maximum height, time of flight, and the impact of gravity on projectile motion. Ideal for students in Grades 9-12.

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