A 400-foot tall l monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 18°, and that the angle of the depression to the bottom of the. monument is 3°. How far is the person from the monument?

To solve this problem, we can use trigonometry and the concept of angles of elevation and depression. Let’s break it down step by step.

### 1. Variables and Setup:
- Height of the monument: \( h = 400 \, \text{ft} \)
- Angle of elevation to the top of the monument: \( \theta_1 = 18^\circ \)
- Angle of depression to the bottom of the monument: \( \theta_2 = 3^\circ \)

Let \( d \) represent the horizontal distance from the person to the base of the monument, and \( h_1 \) represent the height of the person’s position above the base of the monument.

### 2. Trigonometric Equations:
We can establish two right triangles here, one involving the angle of elevation and one involving the angle of depression.

- For the **angle of elevation** to the top of the monument, we use:
  \[
  \tan(\theta_1) = \frac{h - h_1}{d}
  \]
  \[
  \tan(18^\circ) = \frac{400 - h_1}{d}
  \]

- For the **angle of depression** to the bottom of the monument, we use:
  \[
  \tan(\theta_2) = \frac{h_1}{d}
  \]
  \[
  \tan(3^\circ) = \frac{h_1}{d}
  \]

### 3. Solve for \( d \):
From the second equation:
\[
h_1 = d \cdot \tan(3^\circ)
\]

Substitute this into the first equation:
\[
\tan(18^\circ) = \frac{400 - d \cdot \tan(3^\circ)}{d}
\]

Multiply both sides by \( d \):
\[
d \cdot \tan(18^\circ) = 400 - d \cdot \tan(3^\circ)
\]

Now solve for \( d \):
\[
d \cdot (\tan(18^\circ) + \tan(3^\circ)) = 400
\]
\[
d = \frac{400}{\tan(18^\circ) + \tan(3^\circ)}
\]

### 4. Numerical Calculation:
Using the values for the tangents:
\[
\tan(18^\circ) \approx 0.3249, \quad \tan(3^\circ) \approx 0.05241
\]
Substitute these into the equation:
\[
d = \frac{400}{0.3249 + 0.05241} = \frac{400}{0.37731} \approx 1060.16 \, \text{ft}
\]

### Final Answer:
The person is approximately **1060.16 feet** away from the monument.

Would you like further details or clarifications on any step?

---

Here are five related questions:
1. How does the height of the building affect the calculation?
2. How would the result change if the angle of elevation increased?
3. Can you explain why the tangent function was used in both triangles?
4. What would happen if the person was closer to the monument?
5. How would this problem differ if the monument were on a slope?

**Tip:** In trigonometry, the tangent function is commonly used for right triangles because it relates angles to the opposite and adjacent sides.

A 400-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 18°, and that the angle of the depression to the bottom of the monument is 3°. How far is the person from the monument?

Learn how to calculate the distance from a building to a 400-foot tall monument using angles of elevation and depression. This trigonometry problem involves key concepts like tangent functions and right triangles. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation