To solve this problem, we need to perform a quadratic regression using the data points provided in the table, where $x$ represents time in seconds, and $y$ represents height in feet. The general form of a quadratic equation is:

$y = ax^2 + bx + c$

Step 1: Set Up the Data for Quadratic Regression

The data points given are:

• $(0.7, 176)$
• $(1.4, 333)$
• $(1.9, 441)$
• $(2.7, 592)$
• $(3.2, 669)$

Step 2: Perform Quadratic Regression

Using these points, we can calculate the coefficients $a$, $b$, and $c$ for the quadratic equation. I’ll compute these using a regression method to find the best-fit values for $a$, $b$, and $c$.

Let me calculate this.The quadratic regression equation for the data set, with coefficients rounded to the nearest tenth, is:

$y = -17.7x^2 + 267.4x - 3.8$

Step 3: Calculate the Height at $x = 11.8$ Seconds

Now, let's substitute $x = 11.8$ seconds into the equation to find the corresponding height $y$, rounding the result to the nearest foot.The height of the rocket at $x = 11.8$ seconds is approximately 687 feet.

Would you like more details on the steps or any specific part of this calculation?

Here are some related questions to deepen your understanding:

1. How does quadratic regression work, and why do we use it here?
2. What would be the effect of using a linear regression model instead of quadratic for this data?
3. How could you calculate the maximum height of the rocket using the quadratic equation?
4. What are some practical applications of quadratic regression beyond this example?
5. How can we determine the time when the rocket reaches a certain height?

Tip: When performing quadratic regression, it’s essential to verify that a quadratic model is appropriate by examining the data trend; this can prevent overfitting or underfitting.