A ship sails 9 km from point Apapa to Bar Beach on a bearing of $[tex]138^{\circ}$[/tex]. It then changes its course and moves due west until it reaches CMS, which is south of Apapa. Calculate the distance between Bar-Beach and CMS. [Take; $[tex]\operatorname{Sin} 42^{\circ}=0.6991[/tex], $[tex]\operatorname{Cos} 42^{\circ}=0.7431[/tex], $[tex]\tan 42^{\circ}=0.9004[/tex] $[tex]\left.\operatorname{Cos} 48^{\circ}=0.6691\right][/tex]
Answer (2)
The distance between Bar Beach and CMS is approximately 6.29 km, calculated using trigonometric functions based on bearings from Apapa. Using the sine function with the given angles, we determine the length of the segment from Bar Beach to CMS. The ship's journey and course changes were accounted for to arrive at this distance.
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Determine the angle BAC using the bearing: 18 0 ∘ − 13 8 ∘ = 4 2 ∘ .
Apply the sine function in the right-angled triangle ABC: sin ( 4 2 ∘ ) = A B BC .
Substitute the given values: BC = 9 × 0.6991 .
Calculate the distance BC: BC = 6.2919 km. The distance between Bar-Beach and CMS is approximately 6.29 km .
Explanation
Problem Analysis Let's analyze the problem. A ship sails from Apapa to BarBeach, then changes course to CMS, which is directly south of Apapa. We're given the distance from Apapa to BarBeach and the bearing, and we need to find the distance from BarBeach to CMS.
Diagram and Angles Let A represent Apapa, B represent BarBeach, and C represent CMS. The ship sails from A to B, a distance of 9 km, on a bearing of 13 8 ∘ . Then, it sails due west from B to C. Since C is south of A, angle ACB is a right angle. The angle BAC is 18 0 ∘ − 13 8 ∘ = 4 2 ∘ .
Using Sine Function We have a right-angled triangle ABC, where AB = 9 km and ∠ B A C = 4 2 ∘ . We want to find the length BC. In triangle ABC, we can use the sine function: sin ( ∠ B A C ) = A B BC .
Applying Given Value We are given that sin ( 4 2 ∘ ) = 0.6991 . Therefore, BC = A B × sin ( 4 2 ∘ ) = 9 × 0.6991 .
Calculating the Distance Calculating the distance: BC = 9 × 0.6991 = 6.2919 km.
Final Answer The distance between Bar-Beach and CMS is approximately 6.29 km.
Examples
Navigational problems like this are crucial in maritime activities. For instance, a cargo ship needs to travel from one port to another, but due to weather conditions, it has to change its course mid-journey. Calculating the new distance and direction to the final destination becomes essential to ensure the ship arrives safely and on time. This involves using bearings, distances, and trigonometric functions, similar to the problem we solved.
