A ladder 10 meter long is leaning against a vertical wall. If the bottom of the ladder is pulled horizontally away from the wall at the rate of 1.2 meters per seconds, find how fast the top of the ladder is sliding down the wall when the bottom is 6 meters away from the wall

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the problem and set up the diagram
We have a ladder of length 10 meters leaning against a wall. Let:
- \( x \) be the distance from the wall to the bottom of the ladder.
- \( y \) be the height of the top of the ladder from the ground.

### Step 2: Apply the Pythagorean theorem
Since the ladder, wall, and ground form a right triangle, we can use the Pythagorean theorem:
\[ x^2 + y^2 = 10^2 \]
This simplifies to:
\[ x^2 + y^2 = 100 \]

### Step 3: Differentiate with respect to time
We differentiate both sides of the equation with respect to time \( t \):
\[ \frac{d}{dt}(x^2) + \frac{d}{dt}(y^2) = \frac{d}{dt}(100) \]
Using the chain rule, we get:
\[ 2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0 \]
Dividing the entire equation by 2:
\[ x \frac{dx}{dt} + y \frac{dy}{dt} = 0 \]

### Step 4: Solve for \(\frac{dy}{dt}\)
Rearranging the equation gives:
\[ y \frac{dy}{dt} = -x \frac{dx}{dt} \]
Thus, we can express \(\frac{dy}{dt}\):
\[ \frac{dy}{dt} = -\frac{x}{y} \frac{dx}{dt} \]

### Step 5: Substitute known values
We know:
- \( \frac{dx}{dt} = 1.2 \) m/s (the rate at which the bottom of the ladder is pulled away from the wall).
- We need to find \( y \) when \( x = 6 \) m.

Using the Pythagorean theorem:
\[ 6^2 + y^2 = 100 \]
This simplifies to:
\[ 36 + y^2 = 100 \implies y^2 = 64 \implies y = 8 \text{ m} \]

### Step 6: Substitute \( x \), \( y \), and \(\frac{dx}{dt}\) into the equation
Now we can substitute \( x = 6 \), \( y = 8 \), and \( \frac{dx}{dt} = 1.2 \) m/s into the equation for \(\frac{dy}{dt}\):
\[ \frac{dy}{dt} = -\frac{6}{8} \cdot 1.2 \]
Calculating this gives:
\[ \frac{dy}{dt} = -\frac{6 \cdot 1.2}{8} = -\frac{7.2}{8} = -0.9 \text{ m/s} \]

### Conclusion
The negative sign indicates that the top of the ladder is sliding down the wall. Therefore, the top of the ladder is sliding down at a rate of \( 0.9 \) meters per second.

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