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To solve the quadratic regression problem, we need to find a quadratic equation that models the data. The general form of a quadratic equation is:

\[
h(t) = at^2 + bt + c
\]

where:
- \( h(t) \) is the height of the object in meters after \( t \) seconds.
- \( t \) is the time in seconds.
- \( a \), \( b \), and \( c \) are the coefficients to be determined through regression.

The data provided is:

\[
\begin{array}{|c|c|}
\hline
t & h(t) \\
\hline
0.5 & 80.375 \\
1.0 & 93.3 \\
1.5 & 103.775 \\
2.0 & 111.8 \\
2.5 & 117.375 \\
3.0 & 120.5 \\
\hline
\end{array}
\]

Using these data points, we will perform a quadratic regression. I'll calculate the best-fitting quadratic equation for this data.

Let me calculate this for you!The quadratic regression equation that best fits the data is:

\[
h(t) = -4.9t^2 + 33.2t + 65.0
\]

Where:
- \( a = -4.9 \)
- \( b = 33.2 \)
- \( c = 65.0 \)

This equation expresses the height \( h(t) \) of the object as a function of the time \( t \) in seconds after it was thrown.

Would you like more details on this solution or have any specific questions?

Here are 5 related questions you might find helpful:
1. How do you interpret the coefficients of a quadratic equation in real-world scenarios?
2. Can you explain the process of quadratic regression and how it's different from linear regression?
3. What does the vertex of this quadratic equation represent in terms of the object's motion?
4. How would you solve for the time when the object reaches its maximum height using this equation?
5. How would you calculate the total time the object takes to reach the ground using the equation?

**Tip:** When using quadratic equations in real-world situations, the coefficient of \( t^2 \) (here, \( a = -4.9 \)) typically represents acceleration or deceleration in the context of physics (e.g., gravity's effect on a thrown object).

Find the quadratic regression equation. Round coefficients to one decimal place where applicable. Express the height of the object as a function of the number of seconds that have passed since the object was thrown. For example, h(t) = at^2 + bt + c.

Solve the quadratic regression problem to find the equation representing the height of an object over time after it was thrown from a building. The data points provided include heights at specific times, and the quadratic equation is modeled using the form h(t) = at^2 + bt + c. Learn the step-by-step solution for determining the equation.

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