Problem 52 - Algebra and Trigonometry

Question

A boat leaves the entrance to a harbor and travels 40 miles on a bearing of \(\mathrm{S} 64^{\circ} \mathrm{E}\). How many miles south and how many miles east from the harbor has the boat traveled?

Step-by-Step Solution

Answer

The boat has travelled approximately 17 miles south and 36 miles east.

Step 1: Understand the Problem and Draw a Picture

Imagine the problem as a right angle triangle, with the starting point at the harbor. The boat travels at an angle of \(64^{\circ}\) to the south-east, forming the hypotenuse of the right angle triangle. The total distance travelled by the boat (40 miles) will be the hypotenuse of the triangle. Label the sides of the triangle - the side opposite the angle is the 'opposite' and the side adjacent to the angle is the 'adjacent'. The 'opposite' represents the distance travelled east, and the 'adjacent' represents the distance travelled south.

Step 2: Calculate the Distance Traveled South

To find the distance travelled south, you can consider the 'adjacent' side of the triangle. Use the cosine rule here. Cosine of the angle is equal to the adjacent side divided by the hypotenuse. So, mathematically, \(\cos(64^{\circ}) = \frac{Adjacent}{Hypotenuse}\). Solving this for 'Adjacent', we get \(Adjacent = \cos(64^{\circ}) \times Hypotenuse\). Substituting the known values, \(\cos(64^{\circ}) \times 40\) miles, which equals to approximately 17 miles south.

Step 3: Calculate the Distance Traveled East

For the distance travelled east, consider the 'opposite' side. We use the sine rule here. Sine of the angle is equal to opposite side divided by hypotenuse. Hence, \(\sin(64^{\circ}) = \frac{Opposite}{Hypotenuse}\). Solving for 'Opposite', we get \(Opposite = \sin(64^{\circ}) \times Hypotenuse\). Substituting the values, \(\sin(64^{\circ}) \times 40\) miles, which equals to approximately 36 miles east.