To determine the height of the mountain, we can solve this using trigonometry. Here is the step-by-step solution:

Define Variables:

1. Let the height of the mountain be $h$.
2. Let the distance from the first observation point to the base of the mountain be $d$.

From the problem, two angles of elevation are provided:

• At $d$, the angle of elevation is $31^\circ$.
• At $d - 1000$, the angle of elevation is $33^\circ$.

Using the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

We can write two equations for the two observations:

1. From the first point: $\tan(31^\circ) = \frac{h}{d}$ $h = d \cdot \tan(31^\circ)$

2. From the second point: $\tan(33^\circ) = \frac{h}{d - 1000}$ $h = (d - 1000) \cdot \tan(33^\circ)$

Solve for $h$:

Set the two expressions for $h$ equal: $d \cdot \tan(31^\circ) = (d - 1000) \cdot \tan(33^\circ)$

Expand and simplify: $d \cdot \tan(31^\circ) = d \cdot \tan(33^\circ) - 1000 \cdot \tan(33^\circ)$

Group terms with $d$: $d (\tan(31^\circ) - \tan(33^\circ)) = -1000 \cdot \tan(33^\circ)$

Solve for $d$: $d = \frac{-1000 \cdot \tan(33^\circ)}{\tan(31^\circ) - \tan(33^\circ)}$

Calculate $d$:

Using approximate values: $\tan(31^\circ) \approx 0.6009, \quad \tan(33^\circ) \approx 0.6494$ $d = \frac{-1000 \cdot 0.6494}{0.6009 - 0.6494}$ $d = \frac{-649.4}{-0.0485} \approx 13396.9 \, \text{feet}$

Calculate $h$:

Using $h = d \cdot \tan(31^\circ)$: $h = 13396.9 \cdot 0.6009 \approx 8052.1 \, \text{feet}$

Final Answer:

The height of the mountain is approximately 8052 feet.

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Related Questions:

1. How is the tangent function used in real-life surveying problems?
2. What other methods could be used to estimate the height of a mountain?
3. How would the calculation change if the second observation point was farther away instead of closer?
4. How accurate is this estimation if the angles are measured with a small error?
5. What are the implications of atmospheric refraction on angle measurements?

Tip: Always double-check the precision of your angle measurements to ensure accurate results in trigonometric calculations!