To solve the problem, we will determine the height of the building using trigonometry.

Given:

• The surveyor's height = 1.4 m
• Distance from the building = 100 m
• Angle of elevation = 40°

Step 1: Analyze the setup

• The total height of the building is the sum of:

  • The height measured from the surveyor's eyes to the top of the building (denoted as $h$).
  • The height of the surveyor ($1.4 \, \text{m}$).

• Using the tangent function in a right triangle: $\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}$ Here:

  • Opposite side = $h$ (distance from the surveyor's eyes to the top of the building).
  • Adjacent side = $100 \, \text{m}$.

Step 2: Set up the equation

$\tan(40^\circ) = \frac{h}{100}$ Solve for $h$: $h = 100 \cdot \tan(40^\circ)$

Step 3: Calculate $h$

Using $\tan(40^\circ) \approx 0.8391$: $h = 100 \cdot 0.8391 = 83.91 \, \text{m}$

Step 4: Add the surveyor's height

The total height of the building: $\text{Total Height} = h + 1.4 = 83.91 + 1.4 = 85.31 \, \text{m}$

Final Answer:

The height of the building is approximately 85.31 m.

Would you like me to create a sketch or explain further?

Related Questions:

1. How does changing the angle of elevation affect the calculated height?
2. What would happen if the surveyor was closer or farther from the building?
3. How is trigonometry applied in real-life surveying tasks?
4. Can the tangent function be used for angles greater than 90°? Why or why not?
5. How would the calculation change if the ground wasn’t level?

Tip:

Always ensure that your calculator is set to the correct mode (degrees/radians) based on the angle's unit.