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The problem you're asking to solve is:

> "To test the durability of an engineered plastic container, its inventor drops a sample of it from the top of a building. How tall is the building if the container reached the ground at 23.1 m/s in 8 seconds?"

Let's break this down:

1. **Time of fall**: \( t = 8 \) seconds.
2. **Final velocity**: \( v_f = 23.1 \, \text{m/s} \).
3. **Acceleration due to gravity**: \( g = 9.8 \, \text{m/s}^2 \).

We can use the following kinematic equation to find the height \( h \) of the building:

\[
v_f = v_i + g t
\]

Where:
- \( v_f \) is the final velocity,
- \( v_i \) is the initial velocity (which is 0 m/s since it was dropped),
- \( g \) is the acceleration due to gravity,
- \( t \) is the time.

First, we can rearrange this equation to solve for the initial velocity \( v_i \):

\[
23.1 \, \text{m/s} = 0 \, \text{m/s} + (9.8 \, \text{m/s}^2) \times 8 \, \text{seconds}
\]

Let's solve this equation.It seems there was an error in my approach. The initial velocity calculated as \(-55.3 \, \text{m/s}\) doesn't make sense, as it should be zero for an object dropped from rest. Let's correct this by using the correct formula to find the height of the building:

We'll use the following kinematic equation to find the height \( h \):

\[
h = v_i t + \frac{1}{2} g t^2
\]

Since the initial velocity \( v_i = 0 \) (the object was dropped from rest), the equation simplifies to:

\[
h = \frac{1}{2} g t^2
\]

Now, let's calculate the height of the building.The height of the building is **313.6 meters**.

Would you like more details on how this was calculated, or do you have any other questions?

Here are some related questions to consider:

1. How would the height of the building change if the object was thrown downward with an initial velocity?
2. What if the object was thrown upward first—how would you calculate the height?
3. How long would it take for the object to fall if the height of the building were doubled?
4. How does air resistance affect the actual fall time of objects dropped from great heights?
5. If the object was dropped on a planet with a different gravitational acceleration, how would that affect the calculation?

**Tip:** When solving free-fall problems, always ensure you correctly identify whether the object is dropped, thrown upward, or downward, as it changes the initial conditions for your calculations.

Learn how to calculate the height of a building by analyzing the free fall of an object. This problem involves using kinematic equations to find the height based on the final velocity and time of fall. Suitable for students in Grades 9-12.

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