For Question 6 on the image:

The problem appears to involve two poles, $AB$ and $CD$, that are of equal height and standing opposite each other on the ground, which is 60 meters wide. The angles of depression and elevation are given, and we are required to find the height of the pole $CD$.

Given Information:

• The distance between the poles is 60 meters.
• Angle of depression from point $A$ (top of pole $AB$) to point $D$ (bottom of pole $CD$) is $30^\circ$.
• Angle of elevation from point $D$ (bottom of pole $CD$) to point $A$ (top of pole $AB$) is $60^\circ$.

Let's break down the steps to solve the problem using trigonometric relationships.

Step-by-step Solution

1. Label the diagram: Let the height of both poles be $h$.

2. Using the angle of depression (30°):

   • Since the angle of depression from $A$ to $D$ is $30^\circ$, the corresponding angle of elevation from $D$ to $A$ is also $30^\circ$.
   • In triangle $ABD$, we use the tangent formula: $\tan(30^\circ) = \frac{h}{60}$
   • Solving for $h$: $h = 60 \times \tan(30^\circ) = 60 \times \frac{1}{\sqrt{3}} = 60 \times \frac{\sqrt{3}}{3} = 20\sqrt{3} \approx 34.64 \, \text{meters}$

3. Using the angle of elevation (60°):

   • For verification, the angle of elevation from $D$ to $A$ is $60^\circ$, and using the same approach: $\tan(60^\circ) = \frac{h}{60}$
   • Solving for $h$: $h = 60 \times \sqrt{3} \approx 60 \times 1.732 = 103.92 \, \text{meters}$

Conclusion:

Based on both triangles, the calculations verify the consistency of using the given trigonometric values for height determination.

Would you like a detailed step explanation or have any questions about this problem?

Here are 5 related questions to explore further:

1. What are the conditions for using the tangent function in right-angled triangles?
2. How do angles of depression and elevation relate to each other in geometric problems?
3. Can you solve a similar problem using sine or cosine functions instead?
4. What role does the distance between two points play in trigonometric calculations?
5. How would the solution change if the poles had different heights?

Tip: Remember, the tangent of an angle in a right-angled triangle gives the ratio of the opposite side to the adjacent side, which is often useful in height and distance problems.