We can solve this problem using trigonometry and the concept of tangent for right triangles.

Let:

• $h$ = height of the mountain (in feet),
• $x$ = distance from the first observation point to the base of the mountain (in feet).

Step 1: Set up equations using tangent

From the first observation point: $\tan(32^\circ) = \frac{h}{x}$ $h = x \cdot \tan(32^\circ)$

From the second observation point (2000 feet closer): $\tan(34^\circ) = \frac{h}{x - 2000}$ $h = (x - 2000) \cdot \tan(34^\circ)$

Step 2: Equate the two expressions for $h$

$x \cdot \tan(32^\circ) = (x - 2000) \cdot \tan(34^\circ)$

Step 3: Solve for $x$

Expand the right-hand side: $x \cdot \tan(32^\circ) = x \cdot \tan(34^\circ) - 2000 \cdot \tan(34^\circ)$

Rearrange to isolate $x$: $x \cdot (\tan(32^\circ) - \tan(34^\circ)) = -2000 \cdot \tan(34^\circ)$

$x = \frac{-2000 \cdot \tan(34^\circ)}{\tan(32^\circ) - \tan(34^\circ)}$

Step 4: Calculate $x$

Using approximations for tangents:

• $\tan(32^\circ) \approx 0.62487$
• $\tan(34^\circ) \approx 0.67451$

Substitute into the equation: $x = \frac{-2000 \cdot 0.67451}{0.62487 - 0.67451}$ $x = \frac{-1349.02}{-0.04964} \approx 27178.48 \, \text{feet}$

Step 5: Calculate $h$

Using $h = x \cdot \tan(32^\circ)$: $h = 27178.48 \cdot 0.62487 \approx 16978.03 \, \text{feet}$

Final Answer:

The height of the mountain is approximately 16,978 feet.

Would you like more detailed steps, or help with a similar question?

Related Questions:

1. What is the formula for tangent in a right triangle?
2. How does changing the angle of elevation affect the calculation of height?
3. What are the applications of trigonometry in surveying?
4. How accurate is this method in real-life scenarios?
5. Can this be solved graphically instead of algebraically?

Tip:

Always double-check trigonometric values using a calculator for accuracy in applied problems.