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A woman standing on a hill sees a flagpole that she knows is 35 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.)


To find the distance \( x \) from the woman to the flagpole, we can use trigonometric relationships. The problem involves a right triangle with different angles of elevation and depression, which allows us to break the height of the flagpole into two segments and solve for \( x \).

### Step 1: Understand the Triangle Setup
The flagpole height \( h \) is 35 feet. The height can be broken into two parts:
- The height corresponding to the angle of elevation \( \theta_1 = 18^\circ \)
- The height corresponding to the angle of depression \( \theta_2 = 14^\circ \)

### Step 2: Calculate the Components
Let's denote the following:
- \( h_1 \): Height from the bottom of the pole to the observer's eye level (associated with angle of depression).
- \( h_2 \): Height from the top of the pole to the observer's eye level (associated with angle of elevation).
- \( x \): Distance from the observer to the pole.

Given:
- \( h_1 = h \times \sin(14^\circ) \)
- \( h_2 = h \times \sin(18^\circ) \)
- \( h_1 + h_2 = 35 \) ft

### Step 3: Trigonometric Relationships
From the geometry of the situation:
- \( x = \frac{h_1}{\tan(14^\circ)} = \frac{h_2}{\tan(18^\circ)} \)

### Step 4: Solve for \( x \)
Now, we solve for \( x \):

\[
x = \frac{35}{\tan(18^\circ) + \tan(14^\circ)}
\]

Let's calculate the value of \( x \).

### Calculation

I will calculate the exact value of \( x \).The distance \( x \) from the woman to the flagpole is approximately **60.9 feet**.

Would you like more details on how this was calculated or have any other questions?

Here are some related questions you might find interesting:
1. How would the result change if the height of the flagpole was different?
2. What would happen if the angles of elevation and depression were equal?
3. How could we calculate the horizontal distance to the pole if the woman was not on a hill?
4. What trigonometric identities are used to solve such problems?
5. How does the distance change with different angles of elevation or depression?

**Tip:** When solving problems involving angles of elevation and depression, always break down the problem into smaller right triangles to simplify your calculations.

Learn how to calculate the distance from a woman standing on a hill to a 35 ft tall flagpole using trigonometric ratios. Solve for x with detailed step-by-step instructions involving angles of elevation and depression. Suitable for high school students.

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