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The problem asks for the height of a cliff based on the time it takes for a stone to fall and reach the base. Here's a breakdown of how to approach this problem:

1. **Given Information**:
   - The distance fallen by an object varies directly as the square of the time it is in motion.
   - In 3 seconds, the object falls 144.9 feet.
   - The time it takes the stone to reach the base is 2.6 seconds.

2. **Formula**:
   Since the distance fallen \(d\) is proportional to the square of the time \(t\), we can use the equation:
   \[
   d \propto t^2
   \]
   or
   \[
   d = k t^2
   \]
   where \(k\) is the constant of proportionality. From the given information, when \(t = 3\) seconds, \(d = 144.9\) feet. We can first solve for \(k\).

3. **Solving for \(k\)**:
   \[
   144.9 = k (3)^2
   \]
   \[
   144.9 = 9k
   \]
   \[
   k = \frac{144.9}{9} = 16.1
   \]

4. **Finding the height of the cliff**:
   Now, using the value of \(k = 16.1\), we can find the distance fallen when \(t = 2.6\) seconds:
   \[
   d = 16.1 (2.6)^2
   \]
   \[
   d = 16.1 \times 6.76 = 108.836
   \]

5. **Final Answer**:
   The height of the cliff is approximately **108.8 feet**, rounded to one decimal place.

Would you like a more detailed explanation or have any further questions?

Here are 5 related questions to expand your understanding:
1. How does changing the time of fall affect the height from which an object is dropped?
2. What would happen to the proportionality constant \(k\) if the height was given in meters instead of feet?
3. How can air resistance affect the accuracy of this calculation in real-life scenarios?
4. What would be the time required for the object to fall if the height was doubled?
5. How does this formula relate to the basic kinematic equations in physics?

**Tip:** Always ensure you use consistent units (seconds and feet, or seconds and meters) when solving problems involving proportionality or gravity.

The distance that an object falls varies directly as the square of the time the object is in motion. If an object falls for 3 seconds, it will fall 144.9 feet. To estimate the height of a cliff, a person drops a stone at the edge of the cliff and measures how long it takes for the stone to reach the base. If it takes 2.6 seconds, what is the height of the cliff?

Learn how to estimate the height of a cliff using the proportionality between time of fall and distance. This step-by-step solution covers the formula d = k * t^2, where distance varies with the square of time. Example problem: estimating cliff height when a stone takes 2.6 seconds to reach the base. Suitable for Grades 9-12.

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