A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point A on the path, he observes that the angle of elevation of the top of the pillar is `30^0` . After walking for 10 minutes from A in the same direction, at a point B, he observes that the angle of elevation of the top of the pillar is`60^0` . Then the time taken (in minutes) by him, from B to reach the pillar, is : (1) 6 (2) 10 (3) 20 (4) 5

Options

A. 6
B. 10
C. 20
D. 5

Answer

To solve the problem, we will use trigonometric relationships in right triangles formed by the man, the pillar, and the angles of elevation observed at points A and B.

### Step-by-Step Solution:

1. **Define the Variables:**
- Let \( O \) be the top of the pillar.
- Let \( C \) be the base of the pillar.
- Let \( A \) be the initial point where the man observes the angle of elevation of \( 30^\circ \).
- Let \( B \) be the point after walking 10 minutes where the angle of elevation is \( 60^\circ \).
- Let \( OA = Z \) (height of the pillar), \( OB = Y \) (distance from point B to the base of the pillar), and \( OA = X + Y \) (distance from point A to the base of the pillar).

2. **Using Triangle COB (Angle of Elevation at B):**
- From triangle \( COB \):
\[ \tan(60^\circ) = \frac{OC}{OB} \implies \sqrt{3} = \frac{Z}{Y} \implies Z = Y\sqrt{3} \quad \text{(Equation 1)} \]

3. **Using Triangle COA (Angle of Elevation at A):**
- From triangle \( COA \):
\[ \tan(30^\circ) = \frac{OC}{OA} \implies \frac{1}{\sqrt{3}} = \frac{Z}{X + Y} \implies Z = \frac{X + Y}{\sqrt{3}} \quad \text{(Equation 2)} \]

4. **Equating the Two Expressions for Z:**
- From Equation 1 and Equation 2:
\[ Y\sqrt{3} = \frac{X + Y}{\sqrt{3}} \]
- Multiply both sides by \( \sqrt{3} \):
\[ 3Y = X + Y \]
- Rearranging gives:
\[ X = 2Y \]

5. **Finding the Time Taken to Travel Distances:**
- The man takes 10 minutes to walk from \( A \) to \( B \) which corresponds to distance \( X \).
- Since \( X = 2Y \), the time taken to travel distance \( Y \) (from \( B \) to the pillar) is:
\[ \text{Time for } Y = \frac{10 \text{ minutes}}{2} = 5 \text{ minutes} \]

### Conclusion:
The time taken by the man from point B to reach the pillar is **5 minutes**.