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; $\operatorname{Sin} 42^0=0.6991$,
$\begin{array}{l}
\operatorname{Cos} 42^{\circ}=0.7431, \tan 42^{\circ}=0.9004 \\
\left.\operatorname{Cos} 48^{\circ}=0.6691\right]
\end{array}$
A. 3.35 km
B. 4.00 km
C. 5.67 km
D. 6.02 km.
Answer (2)
The ship sails from Apapa to BarBeach on a bearing of 138 degrees, which creates a right triangle where we can calculate the distance to CMS using sine. The calculated distance from BarBeach to CMS is approximately 6.29 km, making D. 6.02 km the closest answer choice provided. Thus, the best choice is option D.
;
Define the locations as points A, B, and C, forming a right-angled triangle.
Calculate the angle BAC using the bearing information: 18 0 0 − 13 8 0 = 4 2 0 .
Apply the sine function to find the distance BC: BC = 9 × sin ( 4 2 0 ) = 9 × 0.6991 = 6.2919 .
Conclude that the distance between Bar-Beach and CMS is approximately 6.29 km.
Explanation
Problem Setup and Diagram Let A be the location of Apapa, B be the location of BarBeach, and C be the location of CMS. The ship sails from A to B on a bearing of 13 8 0 . This means the angle between the north direction at A and the line AB is 13 8 0 . Since CMS is south of Apapa and the ship sails due west from BarBeach to CMS, the line AC is along the south direction and the line BC is along the west direction. Therefore, the angle ACB is 9 0 0 .
Angle Calculation The angle between the south direction at A and the line AB is 18 0 0 − 13 8 0 = 4 2 0 . Therefore, the angle BAC is 4 2 0 . In the right-angled triangle ABC, we know the length of AB is 9 km and the angle BAC is 4 2 0 . We want to find the length of BC.
Applying Sine Function We can use the sine function to relate the angle BAC, the length of AB, and the length of BC: sin ( 4 2 0 ) = A B BC . Therefore, BC = A B ⋅ sin ( 4 2 0 ) = 9 ⋅ sin ( 4 2 0 ) = 9 ⋅ 0.6991 .
Distance Calculation Calculating the value of BC: BC = 9 × 0.6991 = 6.2919 . Therefore, the distance between Bar-Beach and CMS is approximately 6.29 km.
Final Answer The closest answer choice to 6.29 km is D. 6.02 km. However, since we are using sin 4 2 0 = 0.6991 , the exact answer should be 6.2919 km. It seems there might be a slight discrepancy or rounding error in the provided options. However, based on our calculation, the closest answer is approximately 6.29 km.
Examples
Imagine you're sailing a boat and need to navigate to a specific location. By understanding bearings and using trigonometric functions like sine, you can calculate the distances you need to travel in different directions to reach your destination. This is crucial for accurate navigation and avoiding obstacles.
