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.