The rate of increase of length of the shadow of a man 2 metres height, due to a lamp at 10 metres height, when he is moving away from it at the rate of 2m/sec, is

Options

A. `(1)/(2)m//sec`
B. `(2)/(5) m//sec`
C. `(1)/(3)m//sec`
D. `5 m//sec`

Explanation

To solve the problem of finding the rate of increase of the length of the shadow of a man moving away from a lamp, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a lamp of height 10 meters and a man of height 2 meters.
- The man is moving away from the lamp at a rate of 2 meters per second.

2. **Define Variables**:
- Let \( X \) be the distance between the man and the lamp.
- Let \( Y \) be the length of the shadow of the man.

3. **Establish Relationships**:
- From the geometry of the situation, we can form two triangles:
- Triangle \( ECD \) (where \( E \) is the top of the man's head, \( C \) is the base of the lamp, and \( D \) is the tip of the shadow).
- Triangle \( ABD \) (where \( A \) is the top of the lamp, \( B \) is the base of the lamp, and \( D \) is the tip of the shadow).
- Using the tangent function:
- For triangle \( ECD \):
\[ \tan(\theta) = \frac{EC}{CD} = \frac{2}{Y} \]
- For triangle \( ABD \):
\[ \tan(\theta) = \frac{AB}{BD} = \frac{10}{X + Y} \]

4. **Set Up the Equation**:
- Since both triangles share the angle \( \theta \), we can set the two equations equal:
\[ \frac{2}{Y} = \frac{10}{X + Y} \]
- Cross-multiplying gives:
\[ 2(X + Y) = 10Y \]
- Simplifying this:
\[ 2X + 2Y = 10Y \implies 2X - 8Y = 0 \implies X = 4Y \]

5. **Differentiate with Respect to Time**:
- Differentiate both sides with respect to \( t \):
\[ \frac{dX}{dt} = 4 \frac{dY}{dt} \]
- We know from the problem that \( \frac{dX}{dt} = 2 \) m/s (the man is moving away from the lamp).

6. **Substitute Known Values**:
- Substitute \( \frac{dX}{dt} \) into the differentiated equation:
\[ 2 = 4 \frac{dY}{dt} \]

7. **Solve for \( \frac{dY}{dt} \)**:
- Rearranging gives:
\[ \frac{dY}{dt} = \frac{2}{4} = \frac{1}{2} \text{ m/s} \]

### Conclusion:
The rate of increase of the length of the shadow of the man is \( \frac{1}{2} \) meters per second.