To solve this problem, we'll use trigonometric principles, specifically the tangent function. The problem gives us two angles: the angle of depression to the bottom of the flagpole (14°) and the angle of elevation to the top of the flagpole (18°). We'll break the problem into two parts to find the distance $x$ from the woman to the flagpole.

Step 1: Understanding the setup

1. Height of the flagpole = 35 ft.
2. Angle of depression to the bottom of the pole = 14°.
3. Angle of elevation to the top of the pole = 18°.

Step 2: Define variables

Let:

• $x$ be the horizontal distance from the woman to the pole.
• $h$ be the height of the woman above the bottom of the pole.

Step 3: Use trigonometry

For the angle of elevation to the top of the pole:

• We know that $\tan(18^\circ) = \frac{35 + h}{x}$.

For the angle of depression to the bottom of the pole:

• We know that $\tan(14^\circ) = \frac{h}{x}$.

Step 4: Solve for $x$

We can set up two equations from the tangent functions:

1. $\tan(18^\circ) = \frac{35 + h}{x}$ ⟹ $x = \frac{35 + h}{\tan(18^\circ)}$
2. $\tan(14^\circ) = \frac{h}{x}$ ⟹ $h = x \cdot \tan(14^\circ)$

Now, substitute the expression for $h$ from the second equation into the first equation:

$x = \frac{35 + x \cdot \tan(14^\circ)}{\tan(18^\circ)}$

Step 5: Solve for $x$

Let's solve this equation numerically:

$x \cdot \tan(18^\circ) = 35 + x \cdot \tan(14^\circ)$

$x \cdot (\tan(18^\circ) - \tan(14^\circ)) = 35$

$x = \frac{35}{\tan(18^\circ) - \tan(14^\circ)}$

Step 6: Calculate the final answer

Now, calculate the value of $x$ using the tangents of the given angles.

$x = \frac{35}{\tan(18^\circ) - \tan(14^\circ)}$

Let's compute this value:

$x \approx \frac{35}{0.3249 - 0.2493} \approx \frac{35}{0.0756} \approx 462.0 \text{ ft}$

So, the distance $x$ from the woman to the pole is approximately 462.0 feet.

Would you like further details or have any questions?

Here are five related questions to explore:

1. How would the problem change if the height of the flagpole were different?
2. What if the angles of elevation and depression were swapped?
3. How would you solve the problem if only one of the angles was given?
4. Can you derive a general formula for any similar situation?
5. How does the problem change if the flagpole is on a flat surface instead of a hill?

Tip: Always double-check the trigonometric values, as small errors can lead to significantly different results in real-world applications.