A 
100-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 15°
, and that the angle of depression to the bottom of the monument is 3° 
. How far is the person from the monument? Round the answer to the nearest hundredth.

whats the answer in feet?

We can solve this problem using trigonometry by breaking it into two right triangles. Here's how to approach it:

### Step 1: Define the problem.
Let:
- \( d \) be the horizontal distance between the person and the monument.
- \( h_1 = 100 \, \text{feet} \) be the height of the monument.
- \( h_2 \) be the height of the person’s position above the ground in the building.

### Step 2: Use the angle of elevation.
The angle of elevation to the top of the monument is \( 15^\circ \). The tangent of the angle gives the ratio of the height of the monument above the person’s height to the distance \( d \):
\[
\tan(15^\circ) = \frac{h_1 - h_2}{d}.
\]
This simplifies to:
\[
d = \frac{h_1 - h_2}{\tan(15^\circ)}.
\]

### Step 3: Use the angle of depression.
The angle of depression to the bottom of the monument is \( 3^\circ \). This angle gives the ratio of the person’s height to the horizontal distance:
\[
\tan(3^\circ) = \frac{h_2}{d}.
\]
Thus, we can express \( d \) as:
\[
d = \frac{h_2}{\tan(3^\circ)}.
\]

### Step 4: Solve the system of equations.
We now have two equations for \( d \):
1. \( d = \frac{h_1 - h_2}{\tan(15^\circ)} \)
2. \( d = \frac{h_2}{\tan(3^\circ)} \)

Equating these two:
\[
\frac{h_1 - h_2}{\tan(15^\circ)} = \frac{h_2}{\tan(3^\circ)}.
\]

We can now solve for \( h_2 \).

Let me calculate the value of \( h_2 \).The horizontal distance between the person and the monument is approximately 312.15 feet.

Would you like more details or have any further questions?

Here are five related questions:
1. How would the distance change if the angle of elevation was increased?
2. How can you apply the same method if the monument's height was unknown?
3. What role does trigonometry play in navigation or surveying?
4. How would the result change if the person's height above the ground increased?
5. How would you solve the problem if the angles were measured in radians?

**Tip:** When working with angles in trigonometry, always ensure you are using the correct units (degrees or radians) to avoid calculation errors.

A 100-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 15°, and that the angle of depression to the bottom of the monument is 3°. How far is the person from the monument? Round the answer to the nearest hundredth.

Solve a trigonometry problem involving a 100-foot monument by calculating the horizontal distance from a person based on angles of elevation (15°) and depression (3°). Learn the key trigonometric formulas and steps. Suitable for students in Grades 10-12.

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