A ship sails 9 km from point Apapa to BarBeach on a bearing of $138^0$. 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; $\operatorname{Sin} 42^0=0.6991$,
$\begin{array}{l}
\cos 42^{\circ}=0.7431, \tan 42^{\circ}=0.9004 \\
\left.\cos 48^{\circ}=0.6691\right]$
$\end{array}$
Answer (2)
The distance between BarBeach and CMS is approximately 6.2919 km, calculated using the sine function and the established angle. This is derived from the ship's navigation path, following a bearing of 138 degrees. By applying trigonometry, the relationship between distances in the right-angled triangle formed allows us to solve for this distance accurately.
;
Analyzed the problem and visualized the scenario as a right-angled triangle.
Determined the angle at Apapa to be 4 2 0 .
Used the sine function to relate the angle, the distance between Apapa and BarBeach, and the distance between BarBeach and CMS.
Calculated the distance between BarBeach and CMS: 6.2919 km .
Explanation
Problem Analysis Let's analyze the problem. A ship sails from Apapa to BarBeach, covering a distance of 9 km on a bearing of 13 8 0 . Then, it changes course and sails due west from BarBeach to CMS. We know that CMS is located directly south of Apapa. Our goal is to find the distance between BarBeach and CMS.
Visualizing the Problem Imagine a coordinate system where Apapa is at the origin (0,0). The ship's path from Apapa to BarBeach forms an angle of 13 8 0 with the north direction. Since CMS is south of Apapa and the ship travels due west from BarBeach to CMS, we can visualize a right-angled triangle. The vertices of this triangle are Apapa, BarBeach, and CMS, with the right angle at CMS.
Finding the Angle To find the angle inside the right-angled triangle at Apapa, we subtract the bearing from 18 0 0 : 18 0 0 − 13 8 0 = 4 2 0 . This angle is crucial for using trigonometric relationships to find the distance between BarBeach and CMS.
Applying Trigonometry Let's denote the distance between BarBeach and CMS as d 1 . In the right-angled triangle, we can use the sine function to relate the angle at Apapa ( 4 2 0 ), the distance d 1 , and the distance between Apapa and BarBeach (9 km). We have: sin ( 4 2 0 ) = 9 d 1 We are given that sin ( 4 2 0 ) = 0.6991 .
Calculating the Distance Now, we can solve for d 1 :
d 1 = 9 × sin ( 4 2 0 ) = 9 × 0.6991 d 1 = 6.2919 Therefore, the distance between BarBeach and CMS is approximately 6.2919 km.
Final Answer The distance between Bar-Beach and CMS is approximately 6.2919 km.
Examples
Imagine you're navigating a ship and need to determine the distance to a landmark after changing course. This problem demonstrates how bearings and trigonometry can be used to calculate distances in real-world navigation scenarios. By understanding these principles, you can accurately determine your position and plan your route, ensuring a safe and efficient journey. This is also applicable in fields like aviation, surveying, and even robotics, where precise positioning is crucial.
