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Chapter 11: Problem 30

A hiker determines the bearing to a lodge from her current position is \(\mathrm{S} 40^{\circ} \mathrm{W}\). She proceeds to hike 2 miles at a bearing of \(\mathrm{S} 20^{\circ} \mathrm{E}\) at which point she determines the bearing to the lodge is \(\mathrm{S} 75^{\circ} \mathrm{W}\). How far is she from the lodge at this point? Round your answer to the nearest hundredth of a mile.

Short Answer

Expert verified
The distance from the hiker to the lodge is approximately 1.29 miles.

Step by step solution

01

Visualize the Problem

Imagine the hiker's path as a triangle. The initial position is the vertex from which she sees the lodge at \( S 40^{\circ} W \). The path taken is \( S 20^{\circ} E \) for 2 miles and the new position has a bearing of \( S 75^{\circ} W \) to the lodge.
02

Translate Bearing Angles

Convert the bearing angles into angles within the triangle. The bearing \( S 40^{\circ} W \) translates into a triangle angle of \( 40^{\circ} \). Similarly, \( S 20^{\circ} E \) results in an angle \( 20^{\circ} \) from the south line, within the path taken. The new bearing \( S 75^{\circ} W \) translates into a triangle angle of \( 75^{\circ} \).
03

Define the Triangle

Label the vertices: \( A \) is the starting point, \( B \) is the hiker's position after 2-mile hike, and \( C \) is the lodge. \( AB = 2\) miles. The internal angle at \( A \) is \( 180^{\circ} - 20^{\circ} + 40^{\circ} = 100^{\circ} \). Similarly, at \( B \), it \( = 180^{\circ} - 75^{\circ} = 105^{\circ} \). The angle sum rule gives \( \angle C = 75^{\circ} \).
04

Apply the Law of Cosines

To find the distance \( BC \), apply the Law of Cosines: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] where \( a = 2 \) miles, \( b \) is the unknown distance \( BC \), and \( C = 75^{\circ} \). Thus the equation becomes: \[ BC^2 = 2^2 + b^2 - 2 \cdot 2 \cdot b \cdot \cos(75^{\circ}) \].
05

Solve the Equation

Substitute the known values to solve for \( b \): \[ BC^2 = 4 + b^2 - 4b \cdot \cos(75^{\circ}) \]. Using \( \cos(75^{\circ}) \approx 0.2588 \), the equation becomes: \[ BC^2 = 4 + b^2 - 1.0352b \]. Solve this quadratic equation for \( b \) to find \( BC \).
06

Final Calculation and Conclusion

Using a calculator or algebraic techniques gives \( b \approx 1.29 \) after solving the quadratic equation. So, the distance from her position to the lodge is approximately 1.29 miles. Round to two decimal places as needed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bearing Angles
Bearing angles are a method of navigation and direction finding that express the direction of a point in relation to the cardinal directions: North, South, East, and West. These angles are commonly used in activities such as hiking and sailing to convey directions accurately. For instance, a bearing of \( S 40^{\circ} W \) means moving southwards and turning 40 degrees towards the west. Bearings always start from the north or south line and go eastward or westward.
  • Bearing angles simplify complex positions into understandable directions.
  • These angles are always measured clockwise.
  • Knowing how to interpret bearing angles is crucial for navigating from one point to another effectively.
Triangle Geometry
Triangle geometry is essential when dealing with geographic navigation and distance calculation, as real-world paths often form triangles. In the context of our exercise, the hiker's path and position create a triangle between the starting point, current position, and the lodge. Triangles are defined by their vertices (corners) and the angles between them.
  • The sum of angles in any triangle is always \( 180^{\circ} \).
  • To understand the orientation and size of the triangle, convert bearing measurements into internal triangle angles.
  • This requires employing rules such as the angle sum and supplementary angles to calculate missing angles within the triangle.
Distance Calculation
Calculating distances in triangle-based problems usually involves using the Law of Cosines, especially when you know one side and two angles or two sides and an angle. The Law of Cosines relates the lengths of sides of a triangle to the cosine of one of its angles, making it a pivotal tool in navigation and trigonometry.
  • The formula is: \( c^2 = a^2 + b^2 - 2ab \cos(C) \), where \(c\) is the side opposite angle \(C\).
  • This formula is especially useful for calculating unknown distances when other sides and angles are known.
  • Accurate calculations can be achieved by substituting the known values and solving for the unknown side, as seen in the exercise where \( BC \) is found.
Angle Conversion
Angle conversion is the process of translating bearing angles into the internal angles of a triangle for analysis. This step is crucial because it allows for the application of trigonometric principles by giving clear definitions of the triangle's angles.
  • To convert, first determine the angle's reference from North or South.
  • Always subtract the angle from \( 180^{\circ} \) if it crosses one of these reference lines for interior calculations.
  • This conversion enables the practical use of mathematical formulas, linking real-world directions to problem-solving strategies.

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