A surveyor has determined that a mountain is 
h = 2390
 ft high. From the top of the mountain he measures the angles of depression to two landmarks at the base of the mountain and finds them to be 42° and 
c = 44°.
 (Observe that these are the same as the angles of elevation from the landmarks as shown in the figure below.) The angle between the lines of sight to the landmarks is 68°. Calculate the distance between the two landmarks. (Round your answer to the nearest integer.)

We are given the following information about the surveyor's observations:

- Height of the mountain: \( h = 2390 \, \text{ft} \)
- Two angles of depression (or angles of elevation from the landmarks):
  - \( \theta_1 = 42^\circ \)
  - \( \theta_2 = 44^\circ \)
- The angle between the lines of sight to the two landmarks: \( \alpha = 68^\circ \)

To calculate the distance between the two landmarks, we'll use the Law of Cosines and trigonometric relationships.

### Step 1: Determine the distances from the top of the mountain to each landmark

Using the trigonometric relationship for right triangles:

\[
\tan(\theta) = \frac{\text{height}}{\text{distance to landmark}}
\]

Thus, for the two landmarks:

1. For the first landmark (\(d_1\)):
   \[
   \tan(42^\circ) = \frac{2390}{d_1} \quad \Rightarrow \quad d_1 = \frac{2390}{\tan(42^\circ)}
   \]

2. For the second landmark (\(d_2\)):
   \[
   \tan(44^\circ) = \frac{2390}{d_2} \quad \Rightarrow \quad d_2 = \frac{2390}{\tan(44^\circ)}
   \]

### Step 2: Use the Law of Cosines to find the distance between the two landmarks

The Law of Cosines formula is given by:

\[
d^2 = d_1^2 + d_2^2 - 2d_1 d_2 \cos(\alpha)
\]

where \( d \) is the distance between the two landmarks, and \( \alpha = 68^\circ \).

Now, let's calculate this.The distance between the two landmarks is approximately 2872 feet when rounded to the nearest integer.

Would you like any further details or explanations?

Here are five related questions to expand your understanding:
1. How would the result change if one of the angles of elevation was different?
2. How can the Law of Cosines be applied to other real-world problems?
3. What is the significance of the angle between the two lines of sight?
4. How do trigonometric functions relate to the concept of angles of elevation and depression?
5. How could you calculate the height of the mountain if only the distance and angle were provided?

**Tip:** The Law of Cosines is useful for non-right triangles, especially when you know two sides and the angle between them.

A surveyor has determined that a mountain is 2390 ft high. From the top of the mountain he measures the angles of depression to two landmarks at the base of the mountain and finds them to be 42° and 44°. The angle between the lines of sight to the landmarks is 68°. Calculate the distance between the two landmarks. (Round your answer to the nearest integer.)

Learn how to calculate the distance between two landmarks using the height of a mountain and angles of depression. This problem applies trigonometric concepts like the Law of Cosines and right triangle relationships, perfect for high school students in Grades 10-12.

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