Jim is a licensed surveyor who works at a high-rise building project. During the final round of
inspections, he noticed the angle of elevation of the top of a perpendicular building is 25 degrees. Then he moved 110 m nearer to the building and found the angle of the elevation is now 55 degrees. Determine the height of the building at Jim’s worksite (round your answers to two decimal places).

Let's denote the following:

- \( h \) = height of the building
- \( d \) = initial distance from Jim to the building

Given the angles of elevation and the distances moved, we can set up two right triangles and use trigonometric relations to solve for \( h \) and \( d \).

### Step-by-Step Solution:

1. **Initial Setup:**

   When the angle of elevation is 25 degrees:
   \[
   \tan(25^\circ) = \frac{h}{d}
   \]

   When the angle of elevation is 55 degrees after moving 110 meters closer:
   \[
   \tan(55^\circ) = \frac{h}{d - 110}
   \]

2. **Express the height \( h \) in terms of \( d \):**

   From the first equation:
   \[
   h = d \tan(25^\circ)
   \]

3. **Substitute \( h \) into the second equation:**

   \[
   \tan(55^\circ) = \frac{d \tan(25^\circ)}{d - 110}
   \]

4. **Solve for \( d \):**

   \[
   (d - 110) \tan(55^\circ) = d \tan(25^\circ)
   \]

   \[
   d \tan(55^\circ) - 110 \tan(55^\circ) = d \tan(25^\circ)
   \]

   \[
   d (\tan(55^\circ) - \tan(25^\circ)) = 110 \tan(55^\circ)
   \]

   \[
   d = \frac{110 \tan(55^\circ)}{\tan(55^\circ) - \tan(25^\circ)}
   \]

5. **Calculate the values of the tangents:**

   \[
   \tan(25^\circ) \approx 0.4663
   \]
   \[
   \tan(55^\circ) \approx 1.4281
   \]

6. **Substitute these values into the equation for \( d \):**

   \[
   d = \frac{110 \times 1.4281}{1.4281 - 0.4663}
   \]

   \[
   d = \frac{157.091}{0.9618}
   \]

   \[
   d \approx 163.34 \text{ meters}
   \]

7. **Calculate the height \( h \):**

   Using \( h = d \tan(25^\circ) \):

   \[
   h = 163.34 \times 0.4663
   \]

   \[
   h \approx 76.18 \text{ meters}
   \]

### Final Answer:

The height of the building at Jim's worksite is approximately \( 76.18 \) meters.

Would you like more details or have any questions about the solution?

### Relative Questions:
1. How do we derive the tangent function for an angle in a right triangle?
2. What are the general properties of trigonometric functions?
3. Can you explain how the law of sines or cosines might be applied in surveying?
4. What are other methods to measure the height of a building?
5. How do angles of elevation and depression relate to real-world surveying?
6. What role does trigonometry play in navigation and astronomy?
7. How can you use trigonometry to solve real-life problems involving heights and distances?
8. What are the limitations of using trigonometric methods in surveying?

### Tip:
When solving trigonometric problems, always make sure to check the angles and distances to ensure the setup of your equations is accurate.

Learn how to calculate the height of a building using angles of elevation. Jim, a licensed surveyor, uses trigonometric functions and right triangle properties to determine the building's height from his work site. Get a detailed step-by-step solution with angles and distances provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation