From the top of a tower 60 metre high the angle of depression of the top and bottom of a pole are observed to be `45^(@)and60^(@)` respectively .If the pole and tower stand on the same plane ,the height of the pole in metre is

Options

A. a.`60(sqrt(3)-1)`
B. b.`20(sqrt(3)-1)`
C. c.`20(3-sqrt(3))`
D. d.`20(sqrt(3)+1)`

Answer

To solve the problem, we will use trigonometric ratios and properties of right triangles. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a tower (AB) that is 60 meters high.
- The angle of depression to the top of the pole (C) is 45 degrees.
- The angle of depression to the bottom of the pole (D) is 60 degrees.
- We need to find the height of the pole (CD).

2. **Setting Up the Diagram**:
- Let A be the top of the tower, B be the bottom of the tower, C be the top of the pole, and D be the bottom of the pole.
- The height of the tower AB = 60 m.
- The angle of depression from A to C is 45 degrees, and from A to D is 60 degrees.

3. **Using the Angle of Depression**:
- The angle of depression from A to C (top of the pole) is 45 degrees. This means that the angle CAB is also 45 degrees (alternate interior angles).
- The angle of depression from A to D (bottom of the pole) is 60 degrees. Thus, angle DAB is also 60 degrees.

4. **Calculating the Distance from the Tower to the Pole**:
- In triangle ABD (where D is the bottom of the pole):
\[ \tan(60^\circ) = \frac{AB}{BD} \]
\[ \sqrt{3} = \frac{60}{BD} \]
\[ BD = \frac{60}{\sqrt{3}} = 20\sqrt{3} \text{ meters} \]

5. **Calculating the Height of the Pole**:
- In triangle ABC (where C is the top of the pole):
\[ \tan(45^\circ) = \frac{AB}{BC} \]
\[ 1 = \frac{60}{BC} \]
\[ BC = 60 \text{ meters} \]

6. **Finding the Height of the Pole (CD)**:
- Since BD is the distance from the base of the tower to the pole, and BC is the total horizontal distance from the base of the tower to the top of the pole, we can find the height of the pole:
- The height of the pole (CD) can be calculated as:
\[ CD = AB - AE \]
- Here, AE is the height from the bottom of the pole to the top of the pole, which can be found using:
\[ AE = AB - (60 - 20\sqrt{3}) \]
- Therefore:
\[ CD = 60 - 20\sqrt{3} \]

7. **Final Calculation**:
- The height of the pole (CD) is:
\[ CD = 60 - 20\sqrt{3} \text{ meters} \]

### Conclusion:
The height of the pole is \( 60 - 20\sqrt{3} \) meters.