A ship leaves port with a bearing of S 40 W. After traveling 7 miles, the ship turns 90 degrees on a bearing of N 50 W for 11 miles. At that time, what is the bearing of the ship from port?
To determine the bearing of the ship from the port, we can use the concept of vector addition.
Let's break down the ship's movement into two parts:
1. Movement on a bearing of S 40 W for 7 miles.
2. Movement on a bearing of N 50 W for 11 miles.
First, let's represent the initial bearing of S 40 W as a vector. To do this, draw a line segment representing 7 miles in the direction of S 40 W.
Next, represent the second bearing of N 50 W as another vector. Draw a line segment representing 11 miles in the direction of N 50 W, starting from the end of the previous vector.
Now, to find the resultant vector, add the two vectors together by arranging them head to tail.
The resultant vector represents the direct path from the starting point (port) to the ending point (current position of the ship). The bearing of this resultant vector will give us the answer.
To determine the bearing, we can use trigonometry. The bearing is the angle formed between the resultant vector and the reference direction (north).
Let's find the angle. Label the starting point of the first vector as Point A, the ending point of the first vector (and starting point of the second vector) as Point B, and the ending point of the second vector (current position of the ship) as Point C.
Now, we have a triangle ABC.
1. Find the lengths of sides AB and BC using the Pythagorean theorem.
- AB = 7 miles
- BC = 11 miles
2. Use inverse trigonometric functions to find the angles at points A and B in the triangle.
- Angle ABC = atan(AB/BC)
- Angle BAC = atan(BC/AB)
3. Add Angle BAC and the given angle N 50 W to get the bearing of the ship from the port.
So, the bearing of the ship from the port is the sum of the angle formed by the two vectors and the given angle N 50 W.