We appreciate your visit to Mountain A is due west of a walker Mountain B is due north of the walker The guidebook says that Mountain B is 4 3. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
The walker is approximately 3.65 km from Mountain B.
1. Understanding the Setup:
- Mountain A is due west of the walker.
- Mountain B is due north of the walker.
- The straight-line distance between Mountain A and Mountain B is given as 4.3 km.
- The angle between the line connecting Mountain A to the walker and the line connecting Mountain B to the walker is 58°.
2. Identify the Right Triangle Elements:
By visualizing the scenario, you can see that points A (Mountain A), B (Mountain B), and W (Walker) form a right triangle:
- AB (distance between Mountain A and Mountain B) is 4.3 km.
- [tex]\angle WAB[/tex] or the angle at A towards W, is 58°.
- AW (distance from Walker to Mountain A) is adjacent to[tex]\angle WAB[/tex] .
- BW (distance from Walker to Mountain B) is opposite [tex]\angle WAB[/tex] .
3. Finding Distances Using Trigonometric Ratios:
Since the angle given is provided with respect to the right triangle, we use trigonometric functions to determine the lengths of AW and BW.
4. Distance from Walker to Mountains:
- Convert the angle from degrees to radians:
[tex]\[ \text{Angle in radians} = 1.0122909661567112 \text{ rad} \][/tex]
- Calculate the distance from the walker to Mountain A (AW) using cosine:
[tex]\[ \text{distance}_\text{WA} = AB \times \cos(\text{Angle in radians}) = 2.278652836202781 \text{ km} \][/tex]
- Calculate the distance from the walker to Mountain B (BW) using sine:
[tex]\[ \text{distance}_\text{WB} = AB \times \sin(\text{Angle in radians}) = 3.646606813472632 \text{ km} \][/tex]
5. Result:
- The distance from the walker to Mountain A (AW) is approximately 2.28 km.
- The distance from the walker to Mountain B (BW) is approximately 3.65 km.
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