A woman standing on a hill sees a flagpole that she knows is 55 ft tall. The angle of depression to the bottom of the pole is 14°, and the angle of elevation to the top of the pole is 18°. Find her distance *x* from the pole. (Round your answer to one decimal place.)

This problem involves using trigonometry to find the distance from the woman to the flagpole, denoted as \( x \). We'll be using the angles of depression and elevation, along with the height of the flagpole, to set up our equations.

### Step 1: Break the problem into two triangles
1. **Triangle 1**: Involving the angle of depression to the bottom of the pole.
   - The angle of depression is \( 14^\circ \).
   - Let the horizontal distance between the woman and the base of the flagpole be \( x \).
   - The height from the woman's eye level to the base of the pole is unknown (denote this height as \( h_1 \)).

2. **Triangle 2**: Involving the angle of elevation to the top of the pole.
   - The angle of elevation is \( 18^\circ \).
   - The total height of the flagpole is 55 feet, so the height from the woman’s eye level to the top of the pole will be \( 55 - h_1 \).

### Step 2: Set up trigonometric equations
We will use the tangent function because it relates angles to the ratio of the opposite side over the adjacent side.

- For **Triangle 1** (angle of depression):
  \[
  \tan(14^\circ) = \frac{h_1}{x}
  \]
  Therefore:
  \[
  h_1 = x \cdot \tan(14^\circ)
  \]

- For **Triangle 2** (angle of elevation):
  \[
  \tan(18^\circ) = \frac{55 - h_1}{x}
  \]
  Therefore:
  \[
  55 - h_1 = x \cdot \tan(18^\circ)
  \]
  
### Step 3: Solve the system of equations
Substitute \( h_1 = x \cdot \tan(14^\circ) \) into the second equation:
\[
55 - (x \cdot \tan(14^\circ)) = x \cdot \tan(18^\circ)
\]
Simplify and solve for \( x \):
\[
55 = x \cdot (\tan(18^\circ) + \tan(14^\circ))
\]
\[
x = \frac{55}{\tan(18^\circ) + \tan(14^\circ)}
\]

Now, we can calculate \( x \) using the tangent values of \( 18^\circ \) and \( 14^\circ \).

\[
\tan(18^\circ) \approx 0.3249, \quad \tan(14^\circ) \approx 0.2493
\]
\[
x = \frac{55}{0.3249 + 0.2493} = \frac{55}{0.5742}
\]
\[
x \approx 95.8 \, \text{ft}
\]

### Final Answer:
The woman is approximately **95.8 feet** away from the flagpole.

Would you like further details or clarifications?

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Here are 5 related questions you might find helpful:

1. How would the solution change if the flagpole were 70 feet tall?
2. What if the angle of elevation were larger, say 25°?
3. How can the distance be calculated if only one angle is given?
4. Can the method be applied if the observer is standing on a flat surface?
5. How do the angles of elevation and depression affect the height of the triangles?

**Tip:** Always ensure your calculator is set to degrees when dealing with angle problems in trigonometry.

A woman standing on a hill sees a flagpole that she knows is 55 ft tall. The angle of depression to the bottom of the pole is 14°, and the angle of elevation to the top of the pole is 18°. Find her distance x from the pole. (Round your answer to one decimal place.)

Learn how to use trigonometry to find the distance from a woman standing on a hill to a flagpole. This problem involves calculating angles of elevation (18°) and depression (14°), and solving using tangent functions. Suitable for high school students in Grades 10-12.

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