Neglecting air resistance, the distance st) in feet traveled by a freely falling object is given by the function s(t) = 16t^2
, where t is time in seconds. The height of a
certain tower is 805 feet. How long would it take an object to fall to the ground from the top of the building?

To determine how long it would take for an object to fall to the ground from the top of the tower, you can use the given equation for distance traveled by a freely falling object:

\[
s(t) = 16t^2
\]

Here, \(s(t)\) is the distance fallen in feet, and \(t\) is the time in seconds. 

Given:
- The height of the tower is 805 feet, so \(s(t) = 805\).

Now, set up the equation:

\[
805 = 16t^2
\]

Next, solve for \(t\):

1. Divide both sides by 16:

\[
t^2 = \frac{805}{16}
\]

\[
t^2 = 50.3125
\]

2. Take the square root of both sides to solve for \(t\):

\[
t = \sqrt{50.3125}
\]

\[
t \approx 7.1 \text{ seconds}
\]

So, it would take approximately **7.1 seconds** for the object to fall from the top of the building to the ground.

Would you like to go over the steps in more detail or have any questions?

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Here are some related questions you might find interesting:

1. How would the time change if the building were twice as tall?
2. How would the presence of air resistance affect the fall time?
3. What would be the impact of gravity on another planet with a different gravitational constant?
4. Can you derive the formula for \( s(t) = 16t^2 \) from basic principles?
5. How does the initial velocity affect the distance fallen over time?
6. What would happen to the fall time if the object was thrown downward instead of just dropped?
7. How long would it take for an object to fall from a height of 500 feet?
8. How does the fall time relate to the acceleration due to gravity?

**Tip:** When solving quadratic equations, always remember to consider both the positive and negative square roots, but in the context of time, only the positive root makes sense.

Learn how to calculate the time it takes for an object to fall from the top of a tower using the kinematic equation s(t) = 16t^2. This problem involves solving a quadratic equation to determine the time taken for the object to reach the ground, neglecting air resistance.

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