A ladder is resting with the wall at an angle of `30^circ`. A man is ascending the ladder at the rate of 3 ft/sec. His rate of approaching the wall is

Options

A. 3ft/sec
B. `(3)/(2)ft/sec`
C. `(3)/(4)ft/sec`
D. `(3)/(sqrt2)ft/sec`

Explanation

To solve the problem, we need to analyze the situation involving the ladder, the wall, and the man climbing the ladder. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Geometry
The ladder forms a right triangle with the wall and the ground. The angle between the ladder and the wall is given as \(30^\circ\). Therefore, the angle between the ladder and the ground will be \(90^\circ - 30^\circ = 60^\circ\).

### Step 2: Identify the Variables
- The man is ascending the ladder at a rate of \(3 \, \text{ft/sec}\).
- We need to find the rate at which the man is approaching the wall.

### Step 3: Use Trigonometry
To find the rate at which the man is approaching the wall, we can use the cosine component of his velocity. The horizontal component of the man's velocity (the rate at which he approaches the wall) can be found using the cosine of the angle \(60^\circ\).

### Step 4: Calculate the Horizontal Component
The horizontal component of the man's velocity can be calculated as follows:
\[ \text{Horizontal Velocity} = \text{Velocity of Man} \times \cos(60^\circ) \]
Substituting the values:
\[ \text{Horizontal Velocity} = 3 \, \text{ft/sec} \times \cos(60^\circ) \]
Since \(\cos(60^\circ) = \frac{1}{2}\):
\[ \text{Horizontal Velocity} = 3 \, \text{ft/sec} \times \frac{1}{2} = \frac{3}{2} \, \text{ft/sec} \]

### Step 5: Conclusion
The rate at which the man is approaching the wall is \(\frac{3}{2} \, \text{ft/sec}\).

### Final Answer
The rate at which the man is approaching the wall is \(\frac{3}{2} \, \text{ft/sec}\).

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