Problem 58 - Algebra and Trigonometry

Question

A ship leaves port with a bearing of \(\mathrm{S} 40^{\circ} \mathrm{W}\). After traveling 7 miles, the ship turns \(90^{\circ}\) and travels on a bearing of \(\mathrm{N} 50^{\circ} \mathrm{W}\) for 11 miles. At that time, what is the bearing of the ship from port?

Step-by-Step Solution

Answer

The bearing of the ship from the port would be \(\arctan \left(\frac{north}{west} \right)°\) west of north.

Step 1: Drawing the Path

First, draw a diagram to represent the path of the ship. Plot the port, and from there, draw the first vector \(40^{\circ}\) west of due south for 7 miles. From the end of this vector, draw the second vector \(50^{\circ}\) west of due north for 11 miles.

Step 2: Find the Resultant Vector

The resultant vector (from the port to the final location of the ship) can be found by connecting the port and the final location of the ship. We need to find the angle of this vector with respect to the north direction to find the bearing.

Step 3: Calculate the Resultant Vector Components

In order to find the bearing, first the north and east components of each part of the journey need to be found using trigonometry. The components of the first part of the journey (\(40^{\circ}\) W of S and 7 miles) are: South: \(7*\cos(40^{\circ})\), West: \(7*\sin(40^{\circ})\). For the second part of the journey (\(50^{\circ}\) W of N and 11 miles), the components are: North: \(11*\cos(50^{\circ})\), West: \(11*\sin(50^{\circ})\).

Step 4: Summing Up the Components

By adding the north and south components, and the east and west components, we obtain the total north and west components of the resultant vector. The north component is the difference between north and south, while the west component is the sum of the west parts.

Step 5: Find the Bearing

The bearing from the north can be calculated from these components using the arctan function: \( \theta = \arctan \left(\frac{west}{north} \right)\). The actual bearing is obtained by subtracting \(\theta\) from \(90^{\circ}\).