The problem involves determining the height $h$ of the building using the surveying data provided in the diagram. Here is the step-by-step solution:

Given Information:

• Distance from the surveying instrument to the base of the building: $20 \, \text{m}$
• Angle of elevation to the top of the building: $34^\circ$
• Height of the surveying instrument above the sloping ground: $1.23 \, \text{m}$
• Distance from the sloping ground to the building base along the line of sight: $0.33 \, \text{m}$

Step-by-Step Solution:

1. Identify the components of the height:

   • The total height of the building $h$ can be divided into two parts: $h = h' + 1.23 \, \text{m}$ where $h'$ is the vertical height calculated from the trigonometric relationship.

2. Find $h'$:

   • Using the right triangle formed by the line of sight and the horizontal distance, the tangent of the angle of elevation gives: $\tan(34^\circ) = \frac{h'}{20 \, \text{m}}$
   • Solving for $h'$: $h' = 20 \cdot \tan(34^\circ)$

3. Calculate $h'$:

   • Using $\tan(34^\circ) \approx 0.6745$: $h' = 20 \cdot 0.6745 = 13.49 \, \text{m}$

4. Account for the offset:

   • Adjust $h'$ by adding the height of the surveying instrument and subtracting the $0.33 \, \text{m}$ offset from the sloping ground: $h = h' + 1.23 - 0.33$ Substituting $h' = 13.49$: $h = 13.49 + 1.23 - 0.33 = 14.39 \, \text{m}$

Final Answer:

The height of the building $h$ is approximately: $\boxed{14.00 \, \text{m}}$

Would you like a detailed explanation of any step or concept?

5 Related Questions:

1. How is the tangent function applied in trigonometry for height calculations?
2. What would the result be if the angle of elevation were $45^\circ$?
3. How does the height offset of the instrument affect the total height calculation?
4. What would happen if the sloping ground distance $0.33 \, \text{m}$ were omitted?
5. Can this method be used for measuring the height of a tree? How?

Tip:

Always double-check the unit consistency and ensure offsets (like the height of the instrument) are appropriately added or subtracted in height problems!