In this problem, we are given a triangle formed by the flagpole and wall, and we need to calculate the distance $AD$. Let’s break down the information provided:

• The flagpole $AB$ has a height of 5.8 m.
• The horizontal distance $BD$ is 27.3 m.
• The angle of elevation of the top of the wall $B$ from point $D$ is 39°.
• The angle $C$ between $BC$ and $CD$ is 96°.

Step-by-step approach:

1. Right Triangle BCD: Use the information in this triangle to solve for side $BC$. In triangle $BCD$:

   • $BD = 27.3$ m
   • Angle $\angle BDC = 39^\circ$
   • Use the tangent of the angle to solve for $BC$: $\tan(39^\circ) = \frac{BC}{BD}$ Solving for $BC$: $BC = BD \times \tan(39^\circ)$

2. Total distance $AD$: Once $BC$ is known, add the height of the flagpole $AB$ (which is vertical) to $BC$ to find the total distance $AD$.

Let me calculate this now.The total distance $AD$, from point $A$ (the top of the flagpole) to point $D$, is approximately 27.91 meters.

Do you want more details on any of the steps, or do you have any further questions?

Here are 5 related questions you can explore:

1. How would the solution change if the height of the flagpole were different?
2. What if the angle of elevation was larger or smaller—how would it affect the distance?
3. Can we determine the length of $CD$ based on the given angles and distances?
4. How do we apply the Law of Sines or Cosines in problems involving non-right triangles?
5. What other trigonometric identities could simplify this problem?

Tip: Always keep track of the units and make sure they are consistent throughout the problem to avoid calculation errors.