To solve this problem, we need to calculate the height of the building based on the given parameters:

Parameters:

1. Height of the surveyor: $1.4 \, \text{m}$
2. Distance from the building: $100 \, \text{m}$
3. Angle of elevation to the top of the building: $40^\circ$

Solution:

Step 1: Use the tangent formula

The tangent of the angle of elevation relates the opposite side (the height of the building above the surveyor's line of sight) to the adjacent side (distance from the building):

$\tan(\theta) = \frac{\text{height above eye level}}{\text{distance}}$

Here:

• $\theta = 40^\circ$,
• $\text{distance} = 100 \, \text{m}$.

Let the height above the surveyor's eye level be $h_\text{above}$. Rearranging:

$h_\text{above} = \tan(40^\circ) \cdot 100$

Step 2: Calculate $h_\text{above}$

Using $\tan(40^\circ) \approx 0.8391$:

$h_\text{above} = 0.8391 \cdot 100 = 83.91 \, \text{m}$

Step 3: Add the surveyor's height

The total height of the building is:

$\text{Total height} = h_\text{above} + \text{surveyor's height}$

$\text{Total height} = 83.91 + 1.4 = 85.31 \, \text{m}$

Final Answer:

The height of the building is approximately 85.31 meters.

Would you like a detailed sketch or have any additional questions?

Related Questions:

1. What happens if the distance to the building is halved?
2. How would the calculation change if the angle of elevation were $30^\circ$?
3. How can you verify this result using other trigonometric functions?
4. How does the height of the surveyor affect the accuracy of the measurement?
5. What real-life instruments could a surveyor use to measure this angle?

Tip:

Always confirm the angle of elevation is measured from the horizontal and ensure proper units are used in calculations!