Sure, let's work through this problem step-by-step.
### Given:
- Angle measured by Surveyor 1 ([tex]\(\angle_1\)[/tex]): 35 degrees
- Angle measured by Surveyor 2 ([tex]\(\angle_2\)[/tex]): 42 degrees
- Distance between Surveyor 1 and Surveyor 2 ([tex]\(d\)[/tex]): 1200 feet
We want to determine the height of the mountain ([tex]\(h\)[/tex]).
### Steps to solve:
1. Convert angles to radians:
- [tex]\(\angle_1 = 35\)[/tex] degrees [tex]\(= 0.6108652381980153\)[/tex] radians
- [tex]\(\angle_2 = 42\)[/tex] degrees [tex]\(= 0.7330382858376184\)[/tex] radians
2. Calculate the tangents of the angles:
- [tex]\(\tan(\angle_1) = 0.7002075382097097\)[/tex]
- [tex]\(\tan(\angle_2) = 0.9004040442978399\)[/tex]
3. Establish the geometric relationship:
The problem can be visualized as two right triangles sharing a vertical side (the height of the mountain) with their bases along the same horizontal line.
From Surveyor 1:
[tex]\[
\tan(\angle_1) = \frac{h}{b + d}
\][/tex]
From Surveyor 2:
[tex]\[
\tan(\angle_2) = \frac{h}{b}
\][/tex]
Here, [tex]\(b\)[/tex] is the base distance from Surveyor 2 to the projection of the height on the ground.
4. Solve for the base [tex]\(b\)[/tex]:
From Surveyor 2's perspective:
[tex]\[
h = b \cdot \tan(\angle_2)
\][/tex]
From Surveyor 1’s perspective:
[tex]\[
h = (b + d) \cdot \tan(\angle_1)
\][/tex]
Equating the two expressions for [tex]\(h\)[/tex]:
[tex]\[
b \cdot \tan(\angle_2) = (b + d) \cdot \tan(\angle_1)
\][/tex]
[tex]\[
b \cdot \tan(\angle_2) = b \cdot \tan(\angle_1) + d \cdot \tan(\angle_1)
\][/tex]
Rearrange to solve for [tex]\(b\)[/tex]:
[tex]\[
b \cdot (\tan(\angle_2) - \tan(\angle_1)) = d \cdot \tan(\angle_1)
\][/tex]
[tex]\[
b = \frac{d \cdot \tan(\angle_1)}{\tan(\angle_2) - \tan(\angle_1)}
\][/tex]
Substituting in the values:
[tex]\[
b = \frac{1200 \cdot 0.7002075382097097}{0.9004040442978399 - 0.7002075382097097} = -4197.121429690478 \, \text{feet}
\][/tex]
5. Calculate the height [tex]\(h\)[/tex]:
Using [tex]\(h = b \cdot \tan(\angle_2)\)[/tex]:
[tex]\[
h = -4197.121429690478 \cdot 0.9004040442978399 = -3779.1051097024388 \, \text{feet}
\][/tex]
### Height of the mountain:
Thus, the height of the mountain is approximately [tex]\( -3779.11 \)[/tex] feet.
