To solve this problem, we can use trigonometric principles to find the height of the mountain. Let's analyze the problem step-by-step:

Variables:

1. Let $h$ = height of the mountain.
2. Let $x$ = distance from the farther observation point to the base of the mountain.
3. The second observation point is 2000 feet closer to the mountain, so its distance to the base is $x - 2000$.

Trigonometric equations:

From the first observation point: $\tan(25^\circ) = \frac{h}{x}$ $h = x \cdot \tan(25^\circ)$

From the second observation point: $\tan(28^\circ) = \frac{h}{x - 2000}$ $h = (x - 2000) \cdot \tan(28^\circ)$

Equating the two expressions for $h$:

$x \cdot \tan(25^\circ) = (x - 2000) \cdot \tan(28^\circ)$

Solving for $x$:

Expand the equation: $x \cdot \tan(25^\circ) = x \cdot \tan(28^\circ) - 2000 \cdot \tan(28^\circ)$ $x \cdot (\tan(25^\circ) - \tan(28^\circ)) = -2000 \cdot \tan(28^\circ)$ $x = \frac{2000 \cdot \tan(28^\circ)}{\tan(28^\circ) - \tan(25^\circ)}$

Substituting trigonometric values:

$\tan(25^\circ) \approx 0.4663,\ \tan(28^\circ) \approx 0.5317$ $x = \frac{2000 \cdot 0.5317}{0.5317 - 0.4663}$ $x = \frac{1063.4}{0.0654} \approx 16259.45 \text{ feet}$

Finding $h$:

Using $h = x \cdot \tan(25^\circ)$: $h = 16259.45 \cdot 0.4663 \approx 7579.7 \text{ feet}$

Final Answer:

The height of the mountain is approximately 7579.7 feet.

Do you want more detailed steps, or do you have any questions?

Related Questions:

1. How can this method be adapted for non-level terrain?
2. What would happen if the distance between observation points was unknown?
3. Can this problem be solved graphically or using software tools like GeoGebra?
4. How accurate is the result if the angles are measured with slight errors?
5. What are practical ways survey teams ensure precise angle measurements?

Tip:

Always double-check the angle measurements and ensure they are taken from the same height level for accuracy in trigonometric calculations.