Question

What steps would you use to solve the following problem? Explain how you would figure out what the key terms are.

You are standing 100 feet away from a very tall building. Your line of sight is 6 feet from the ground. If the angle of elevation ( Measured by a protractor ) to the top of the building is 80 degrees, how tall is the building?

Answer 1

To solve the problem of finding the height of the building when given the distance from the building, the height of your line of sight, and the angle of elevation, you can follow these steps:

1. **Identify Key Terms**:
- **Distance**: You are standing 100 feet away from the building.
- **Line of Sight Height**: Your line of sight is 6 feet above the ground.
- **Angle of Elevation**: The angle from your line of sight (6 feet) to the top of the building is 80 degrees.
- **Height of the Building**: This is the unknown we want to find.

The key terms that need to be clarified for solving this problem are:
- The distance you are from the building (horizontal distance).
- The height at which your eye level is (6 feet).
- The angle of elevation (80 degrees).
- The height from the ground to the top of the building, which we will denote as \(h\).

2. **Draw a Diagram**:
- It helps to visualize the problem. Draw a right triangle where one vertex is at your eyes, one is at the top of the building, and the last one is on the ground at the base of the building.
- Label the distance from you to the building (100 feet) as the horizontal leg of the triangle, the distance from your line of sight to the top of the building as the vertical leg (which will be \(h - 6\), since you are already 6 feet above the ground), and the angle of elevation as 80 degrees.

3. **Set Up the Relationship**:
- Use the tangent function, which relates the angle of elevation to the opposite and adjacent sides of a right triangle:
\[
\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}}
\]
In our case:
\[
\tan(80^\circ) = \frac{h - 6}{100}
\]

4. **Solve for the Height**:
- Rearrange the equation to isolate \(h\):
\[
h - 6 = 100 \tan(80^\circ)
\]
\[
h = 100 \tan(80^\circ) + 6
\]

5. **Calculate the Height**:
- Use a calculator to find \(\tan(80^\circ)\) and then compute \(h\):
\[
h \approx 100 \cdot \tan(80^\circ) + 6
\]
- Make sure your calculator is set to degrees for the correct calculation of the tangent.

6. **Final Answer**:
- Compute \(h\) to get the total height of the building.

By following these steps methodically, you can find the height of the building based on the provided measurements and angle.