The problem you're asking about involves a 60 m high tower, and the angles of depression to the top and bottom of a pillar are given as 30° and 60°, respectively. The goal is to find the height of the pillar.

We can solve this using trigonometry, specifically by applying the concept of right triangles and the tangent function.

Let’s break it down:

1. Diagram Setup:

   • Let the height of the pillar be $h$ meters.
   • The horizontal distance between the base of the tower and the base of the pillar is $d$ meters.
   • The top of the tower is 60 m above the ground.
   • The angle of depression to the top of the pillar is 30°.
   • The angle of depression to the bottom of the pillar is 60°.

2. Using the Tangent Function:

   • From the top of the tower, using the angle of depression 30° to the top of the pillar, we have: $\tan(30^\circ) = \frac{60 - h}{d}$
   • Similarly, using the angle of depression 60° to the bottom of the pillar, we have: $\tan(60^\circ) = \frac{60}{d}$

3. Values of Tangent:

   • We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}}$ and $\tan(60^\circ) = \sqrt{3}$.

4. Set Up Equations:

   • From $\tan(30^\circ) = \frac{1}{\sqrt{3}}$: $\frac{60 - h}{d} = \frac{1}{\sqrt{3}}$ So, $60 - h = \frac{d}{\sqrt{3}}$. (Equation 1)

   • From $\tan(60^\circ) = \sqrt{3}$: $\frac{60}{d} = \sqrt{3}$ So, $d = \frac{60}{\sqrt{3}} = 20\sqrt{3}$. (Equation 2)

5. Substitute $d$ in Equation 1:

   • Substitute $d = 20\sqrt{3}$ into Equation 1: $60 - h = \frac{20\sqrt{3}}{\sqrt{3}} = 20$ So, $h = 60 - 20 = 40$.

Thus, the height of the pillar is 40 meters.

Final Answer:

The correct option is (c) 40 m.

Do you need further clarification or more details? Here are some related questions to deepen your understanding:

1. How is the tangent function used to solve problems involving right triangles?
2. Can you explain why the angle of depression equals the angle of elevation in these types of problems?
3. What would happen to the solution if the height of the tower changed?
4. How would you solve this problem if the angles of depression were different?
5. What other trigonometric functions can be used for problems involving heights and distances?

Tip: Always draw a diagram for problems involving angles of elevation or depression. It helps visualize the relationships between the angles and distances.