https://file.aiquickdraw.com/mathbot/file/17273091998064g5rifkb.jpg

The problem presented involves using the table of data to generate a quadratic regression that models the height \( h(t) \) of an object over time \( t \), after being thrown from a building.

### Given Table:
| \( t \) (s) | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
|---|---|---|---|---|---|---|
| \( h \) (m) | 67.925 | 77.4 | 84.425 | 89 | 91.125 | 90.8 |

The task involves calculating a quadratic regression equation for \( h(t) \) in the form:
\[
h(t) = at^2 + bt + c
\]
The quadratic regression equation provided in the image is:
\[
h(t) = -2t^2 + 16.5t + 59.9
\]
This equation is rounded to one decimal place, as specified.

#### Interpretation:
1. The quadratic coefficient \( a = -2 \) indicates a downward-opening parabola, implying that the object rises initially, then starts falling after a certain point.
2. The linear coefficient \( b = 16.5 \) reflects the object's initial velocity after being thrown.
3. The constant \( c = 59.9 \) represents the object's initial height at \( t = 0 \).

Let me know if you'd like further details, such as how quadratic regression is calculated, or any additional steps you need help with.

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Would you like further clarification or details?

Here are 5 questions to expand on the topic:
1. How does the quadratic model relate to the object's motion?
2. What is the significance of the negative coefficient for \( t^2 \)?
3. Can you use this model to determine the maximum height reached by the object?
4. How would you solve for the time when the object reaches the ground?
5. How does rounding affect the accuracy of the regression model?

**Tip:** In projectile motion problems, the quadratic term typically comes from the influence of gravity on the object's motion.

The table shows the height of an object thrown off the top of a building as a function of time, and a quadratic regression model is required to express the height of the object over time.

Learn how to calculate a quadratic regression model for the height of an object thrown from a building over time, with example values including t (seconds) and h (meters). Quadratic regression results in the equation h(t) = -2t^2 + 16.5t + 59.9.

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