https://file.aiquickdraw.com/mathbot/file/1730946177135pqghlu9n.jpg

To tackle this problem, I'll approach it in two main steps:

1. **Quadratic Regression**: Fit a quadratic model of the form \( y = ax^2 + bx + c \) using the given data points. I will calculate the coefficients \( a \), \( b \), and \( c \), rounding them to the nearest hundredth as requested.

2. **Prediction for \( x = 10.1 \)**: Once we have the quadratic equation, I’ll use it to find the height at \( x = 10.1 \) seconds.

I'll start with the regression calculations based on the data provided:

- **Data Points**:
    - \( (1.1, 290) \)
    - \( (2.2, 548) \)
    - \( (3.3, 755) \)
    - \( (4.5, 943) \)
    - \( (5.6, 1086) \)
    - \( (6.5, 1154) \)

Let me calculate this for you.The quadratic regression equation for the height \( y \) in terms of time \( x \) is:

\[
y = -16.23x^2 + 283.75x - 1.51
\]

Using this equation, the height of the rocket at \( x = 10.1 \) seconds is approximately **1209 feet**.

Would you like further details on the calculation process, or do you have any questions?

Here are some related questions to expand on this topic:
1. How can we verify the accuracy of a quadratic regression model?
2. What does each coefficient in a quadratic equation represent in terms of the rocket's motion?
3. Could a cubic model provide a better fit for this data, and how would it change the height prediction?
4. How can we interpret the vertex of the quadratic equation in the context of this rocket's trajectory?
5. What are some potential real-world applications of quadratic regression beyond rocket trajectories?

**Tip**: Quadratic regression is especially useful for modeling situations where there is a maximum or minimum value, as it captures parabolic relationships effectively.

A rocket is shot off from a launcher. The accompanying table represents the height of the rocket at given times, where x is time, in seconds, and y is height, in feet. Write a quadratic regression equation for this set of data, rounding all coefficients to the nearest hundredth. Using this equation, find the height, to the nearest foot, at a time of 10.1 seconds.

Learn how to perform a quadratic regression to model the height of a rocket over time. Using data points such as (1.1, 290), (2.2, 548), and others, we derive a quadratic equation to predict the rocket's height at 10.1 seconds. This example demonstrates essential concepts in algebra and regression analysis.

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