The top of a ladder 6 metres long is resting against a vertical wall on a level pavement, when the ladder begins to slide outwards. At the moment when the foot of the ladder is 4 metres from the wall, it is sliding away from the wall at the rate of 0.5 m/sec. How fast is the top-sliding downwards at this instance? How far is the foot from the wall when it and the top are moving at the same rate?

Explanation

To solve the problem, we will use the Pythagorean theorem and implicit differentiation. Here’s a step-by-step solution:

### Step 1: Understand the setup
We have a right triangle formed by the ladder, the wall, and the ground. Let:
- \( x \) = distance from the foot of the ladder to the wall
- \( y \) = height of the top of the ladder from the ground
- The length of the ladder is constant at 6 meters.

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