A ladder is inclined to a wall making an angle of 30° with it. A man is ascending the ladder at the rate of 3 m/sec. How fast is he approaching the wall ?

Answer

To solve the problem, we will analyze the situation using trigonometric relationships and the concept of components of velocity.

### Step 1: Understand the Setup
We have a ladder inclined against a wall at an angle of 30°. A man is ascending the ladder at a speed of 3 m/sec. We need to find out how fast he is approaching the wall.

### Step 2: Identify the Angles
Since the ladder makes an angle of 30° with the wall, the angle between the ladder and the ground is 90° - 30° = 60°.

### Step 3: Break Down the Velocity
The man is ascending the ladder at a speed of 3 m/sec. We can break this velocity into two components:
1. The component of velocity towards the wall (horizontal component).
2. The component of velocity perpendicular to the wall (vertical component).

### Step 4: Calculate the Horizontal Component
The horizontal component of the man's velocity (the rate at which he is approaching the wall) can be calculated using the cosine of the angle between the ladder and the wall.

\[ \text{Horizontal Component} = \text{Speed} \times \cos(60°) \]

Given that the speed of the man is 3 m/sec:

\[ \text{Horizontal Component} = 3 \times \cos(60°) \]

### Step 5: Substitute the Value of Cosine
We know that \(\cos(60°) = \frac{1}{2}\):

\[ \text{Horizontal Component} = 3 \times \frac{1}{2} = 1.5 \text{ m/sec} \]

### Conclusion
The man is approaching the wall at a speed of 1.5 m/sec.

### Summary of Steps:
1. Identify the angles in the setup.
2. Break down the man's velocity into components.
3. Use trigonometric functions to calculate the horizontal component of the velocity.
4. Substitute the known values and calculate the result.