A 200-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 
14 degrees, and that the angle of depression to the bottom of the monument is 
3 degrees. How far is the person from the monument? Round the answer to the nearest hundredth.

To solve this problem, we can use trigonometry, specifically the tangent function, which relates an angle in a right triangle to the opposite side and the adjacent side.

### Step 1: Break down the problem

We have two angles:
- The angle of elevation to the top of the monument is 14 degrees.
- The angle of depression to the bottom of the monument is 3 degrees.

Let \( d \) be the horizontal distance from the person to the monument, and let the height of the person above the ground be \( h \).

We will solve for \( d \) using the tangent function.

### Step 2: Set up two right triangles

We can split the height of the monument (200 feet) into two parts:
1. The height from the person's eye level to the top of the monument.
2. The height from the person's eye level to the base of the monument.

Let \( h_1 \) represent the height from the person's eye level to the top of the monument, and \( h_2 \) represent the height from the person's eye level to the bottom of the monument.

Thus, the height of the monument can be written as:
\[
200 = h_1 + h_2
\]

Using the tangent function:
1. For the angle of elevation to the top of the monument:
\[
\tan(14^\circ) = \frac{h_1}{d}
\]
So, \( h_1 = d \cdot \tan(14^\circ) \).

2. For the angle of depression to the bottom of the monument:
\[
\tan(3^\circ) = \frac{h_2}{d}
\]
So, \( h_2 = d \cdot \tan(3^\circ) \).

### Step 3: Solve for \( d \)

Now, using the fact that \( h_1 + h_2 = 200 \), we substitute the expressions for \( h_1 \) and \( h_2 \) into the equation:
\[
d \cdot \tan(14^\circ) + d \cdot \tan(3^\circ) = 200
\]
Factor out \( d \):
\[
d \cdot (\tan(14^\circ) + \tan(3^\circ)) = 200
\]
Solve for \( d \):
\[
d = \frac{200}{\tan(14^\circ) + \tan(3^\circ)}
\]

### Step 4: Calculate the value of \( d \)

Now, plug in the values of the tangent of 14 degrees and 3 degrees:

\[
d = \frac{200}{\tan(14^\circ) + \tan(3^\circ)} \approx \frac{200}{0.2493 + 0.0524} = \frac{200}{0.3017} \approx 663.03
\]

So, the horizontal distance from the person to the monument is approximately **663.03 feet**.

### Would you like further details or clarifications?

#### Related Questions:
1. How do tangent and other trigonometric functions relate to right triangles?
2. What is the difference between the angle of elevation and the angle of depression?
3. How do we use trigonometry in real-life scenarios like this one?
4. What if the height of the person above the ground is given? How would it change the solution?
5. How does rounding affect the accuracy of the final answer?

#### Tip:
Always ensure that your calculator is in degree mode when working with angles in degrees!

A 200-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 14 degrees, and that the angle of depression to the bottom of the monument is 3 degrees. How far is the person from the monument? Round the answer to the nearest hundredth.

Learn how to calculate the distance from a person to a 200-foot monument using trigonometric concepts like angles of elevation and depression. This problem uses the tangent function to find the horizontal distance, applying real-life trigonometry for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation