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 find how far the person is from the monument, we can use trigonometric relationships based on the angles of elevation and depression.

### Step 1: Define variables
Let:
- \( d \) be the horizontal distance from the person to the monument.
- \( h \) be the height of the building window from which the person is viewing the monument.
- The monument's height is given as 400 feet.

The angle of elevation to the top of the monument is 18°, and the angle of depression to the bottom of the monument is 3°.

### Step 2: Use trigonometry to set up equations
We can break this problem into two right triangles:
1. **Top of the monument**: The angle of elevation to the top of the monument is 18°, and the height difference between the person and the top of the monument is \( 400 - h \).
2. **Bottom of the monument**: The angle of depression to the bottom of the monument is 3°, and the height difference between the person and the ground (bottom of the monument) is \( h \).

Using the tangent function:
- For the top triangle:
  \[
  \tan(18^\circ) = \frac{400 - h}{d}
  \]
  Rearranging, we get:
  \[
  d = \frac{400 - h}{\tan(18^\circ)}
  \]

- For the bottom triangle:
  \[
  \tan(3^\circ) = \frac{h}{d}
  \]
  Rearranging, we get:
  \[
  d = \frac{h}{\tan(3^\circ)}
  \]

### Step 3: Set up an equation
Since both expressions represent the same distance \( d \), we can set them equal to each other:
\[
\frac{400 - h}{\tan(18^\circ)} = \frac{h}{\tan(3^\circ)}
\]

### Step 4: Solve for \( h \)
Now, solve this equation for \( h \):
\[
(400 - h)\tan(3^\circ) = h\tan(18^\circ)
\]
Expanding:
\[
400\tan(3^\circ) - h\tan(3^\circ) = h\tan(18^\circ)
\]
Move terms involving \( h \) to one side:
\[
400\tan(3^\circ) = h(\tan(18^\circ) + \tan(3^\circ))
\]
Solve for \( h \):
\[
h = \frac{400\tan(3^\circ)}{\tan(18^\circ) + \tan(3^\circ)}
\]

### Step 5: Calculate values
Using a calculator:
- \( \tan(18^\circ) \approx 0.3249 \)
- \( \tan(3^\circ) \approx 0.05241 \)

Substitute these values into the equation:
\[
h = \frac{400 \times 0.05241}{0.3249 + 0.05241} \approx \frac{20.964}{0.37731} \approx 55.56 \, \text{feet}
\]

### Step 6: Find \( d \)
Now that we know \( h \), we can substitute it back into the expression for \( d \):
\[
d = \frac{55.56}{\tan(3^\circ)} \approx \frac{55.56}{0.05241} \approx 1,059.76 \, \text{feet}
\]

### Final Answer:
The person is approximately **1,059.76 feet** from the monument.

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Would you like further clarification on any of the steps or details?

Here are some related questions:
1. What if the height of the monument were different? How would the distance change?
2. How would the calculations change if the angles of elevation and depression were reversed?
3. Can we apply the same principles to calculate the height of the building?
4. How does changing the angle of elevation affect the horizontal distance?
5. What is the relationship between the tangent function and right triangles in this context?

**Tip:** Always double-check your trigonometric functions and ensure your calculator is in degree mode when working with angles in degrees!

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 depression to the bottom of the monument is 3°. How far is the person from the monument?

Learn how to calculate the distance between a person and a 400-foot tall monument using angles of elevation and depression. This trigonometry problem uses key formulas like tangent to solve for distance. Suitable for high school students (Grades 9-12).

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