To solve this problem, we can use trigonometry, specifically the tangent function, since it relates the angle of elevation to the height of the mountain and the distance from the observation point to the mountain.

Given information:

1. First observation:

   • Angle of elevation, $\theta_1 = 24^\circ$
   • Distance from the mountain = $d$ (unknown)

2. Second observation:

   • Angle of elevation, $\theta_2 = 28^\circ$
   • Distance from the mountain is $d - 1000$ feet.

3. The height of the mountain = $h$ (unknown, to be found).

Steps to solve:

1. Set up equations based on the tangent function. For the first observation point: $\tan(24^\circ) = \frac{h}{d}$ So, $h = d \cdot \tan(24^\circ)$.

2. For the second observation point (closer by 1000 feet): $\tan(28^\circ) = \frac{h}{d - 1000}$ So, $h = (d - 1000) \cdot \tan(28^\circ)$.

3. Set the equations for $h$ equal to each other: $d \cdot \tan(24^\circ) = (d - 1000) \cdot \tan(28^\circ)$

4. Solve for $d$: Expanding and rearranging terms, we get: $d \cdot \tan(24^\circ) = d \cdot \tan(28^\circ) - 1000 \cdot \tan(28^\circ)$ $d \cdot (\tan(24^\circ) - \tan(28^\circ)) = -1000 \cdot \tan(28^\circ)$ $d = \frac{1000 \cdot \tan(28^\circ)}{\tan(28^\circ) - \tan(24^\circ)}$

5. Substitute the values of the tangents:

   • $\tan(24^\circ) \approx 0.4452$
   • $\tan(28^\circ) \approx 0.5317$

   $d = \frac{1000 \cdot 0.5317}{0.5317 - 0.4452}$ $d \approx \frac{531.7}{0.0865} \approx 6148.84 \text{ feet}$

6. Calculate the height $h$ using $d$: $h = d \cdot \tan(24^\circ)$ $h \approx 6148.84 \cdot 0.4452 \approx 2737.08 \text{ feet}$

Answer:

The height of the mountain is approximately 2737.08 feet, rounded to two decimal places.

Would you like further details on any step?

Related Questions:

1. How would the answer change if the second observation point was only 500 feet closer?
2. What is the effect of increasing the angle of elevation in these calculations?
3. Can this method be used for any two arbitrary angles? Why or why not?
4. How would you set up a similar problem if you only had the angle from one observation point?
5. Why does the tangent function work well in this problem?

Tip:

Always double-check trigonometric calculations when using small angle differences, as they can significantly impact the result.