- Scenario: A 25-foot ladder is leaning against a wall. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. How fast is the top of the ladder sliding down the wall when the base is 15 feet from the wall

To solve this problem, we can use related rates and the Pythagorean theorem. 

1. **Identify Variables:**
   - Let \( x \) be the distance from the base of the ladder to the wall.
   - Let \( y \) be the height of the top of the ladder on the wall.
   - The length of the ladder is a constant \( L = 25 \) feet.

2. **Pythagorean Theorem:**
   \[
   x^2 + y^2 = L^2
   \]
   \[
   x^2 + y^2 = 25^2 = 625
   \]

3. **Differentiate with respect to time \( t \):**
   \[
   2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0
   \]

4. **Solve for \( \frac{dy}{dt} \):**
   \[
   \frac{dy}{dt} = -\frac{x}{y} \cdot \frac{dx}{dt}
   \]

5. **Given Values:**
   - \( \frac{dx}{dt} = 2 \) feet/second (the rate at which the base is pulled away).
   - \( x = 15 \) feet (the distance from the wall).

6. **Find \( y \) when \( x = 15 \):**
   \[
   15^2 + y^2 = 625
   \]
   \[
   225 + y^2 = 625
   \]
   \[
   y^2 = 400 \implies y = 20 \text{ feet}
   \]

7. **Substituting into the rate equation:**
   \[
   \frac{dy}{dt} = -\frac{15}{20} \cdot 2 = -\frac{15}{10} = -1.5 \text{ feet/second}
   \]

### Conclusion
The top of the ladder is sliding down the wall at a rate of **1.5 feet per second** when the base is 15 feet from the wall.

Would you like more details on any part of this solution? Here are some related questions you might consider:

1. How would the rate change if the ladder were longer or shorter?
2. What is the angle of the ladder with the ground at that moment?
3. How does this scenario apply in real-life situations (like firefighting)?
4. What happens when the base is pulled away at different rates?
5. Can you visualize this scenario using a graph?

**Tip:** When solving related rates problems, always start by identifying relationships between variables and write down what you know first!

Scenario: A 25-foot ladder is leaning against a wall. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. How fast is the top of the ladder sliding down the wall when the base is 15 feet from the wall?

Explore a related rates problem involving a 25-foot ladder leaning against a wall, with the base moving away at 2 feet/second. Discover how fast the top slides down when the base is 15 feet from the wall, applying the Pythagorean theorem and differentiation techniques. Ideal for Grades 11-12 students.

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