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This problem describes a quadratic regression model for a rocket's height as a function of time. The question asks for the time \( t \) when the rocket will land back on the ground, meaning the height will be zero.

We can extract the following data points from the problem:
- At \( t = 0 \), height \( h = 0 \) meters (launch point).
- At \( t = 1 \), height \( h = 20.1 \) meters.
- At \( t = 2 \), height \( h = 30.4 \) meters.

Since the height is modeled by a quadratic equation of the form:
\[
h(t) = at^2 + bt + c
\]
we can use the three points to solve for the coefficients \( a \), \( b \), and \( c \).

Let me calculate the coefficients and solve for the time \( t \) when the rocket lands (i.e., when \( h(t) = 0 \)).The quadratic equation modeling the rocket's height is:

\[
h(t) = -4.9t^2 + 25t
\]

Solving for when \( h(t) = 0 \), the solutions are:
- \( t = 0 \) seconds (the time of launch),
- \( t \approx 5.102 \) seconds.

Thus, the rocket will land back on the ground at \( t \approx 5.102 \) seconds. The correct answer is **D**.

Would you like more details or have any other questions?

Here are 5 related questions to consider:
1. How is the quadratic model derived from the data points?
2. What physical principles influence the shape of the rocket's height graph?
3. How do we interpret the coefficient \( a = -4.9 \) in the equation?
4. What happens to the rocket’s velocity as it reaches its peak height?
5. How can you verify the time it takes for the rocket to reach its highest point?

**Tip:** The coefficient of \( t^2 \) in such projectile motion problems often corresponds to half the acceleration due to gravity (e.g., \(-9.8/2 = -4.9\)).

A model rocket is launched from a site on level ground. One second after the rocket is launched, it is at a height of 20.1 meters above the ground. One additional second later, it is at a height of 30.4 meters above the ground. A quadratic regression using the heights of the rocket at times t = 0, t = 1, and t = 2 seconds is calculated to model the rocket’s height above the ground, in meters, at time t seconds. At what time t seconds does the model predict that the rocket will land back on the ground?

Learn how to model the height of a rocket using quadratic regression and solve for the time it will land. This problem involves calculating the coefficients of a quadratic equation based on given heights and solving for when the height equals zero.

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