Aship leaves its home port expecting to travel to a port 400 km due south. Before it moves even 1 km, a severe storm blows it 150 km due east. In what direction must it travel to reach its destination?
measuring angles from S to W,
TanTheta=150/400
It travels in the direction of Theta, W of S
2. A ship leaves port on a course of and travels 160 miles. Then the ship changes course to a bearing of and travels another 220 miles to its destination.
To determine the direction the ship must travel to reach its destination, we can use vector addition. We need to find the resultant vector by considering the displacement due south and the displacement due east caused by the storm.
Let's represent the displacement due south as a vector with a magnitude of 400 km pointing downward. Similarly, the displacement due east caused by the storm can be represented as a vector with a magnitude of 150 km pointing to the right.
To find the resultant vector, we need to add these two vectors. We can use the Pythagorean theorem to find the magnitude of the resultant vector and the tangent function to find the direction.
The magnitude of the resultant vector is given by the formula:
resultant_magnitude = sqrt((magnitude_south)^2 + (magnitude_east)^2)
In this case:
resultant_magnitude = sqrt(400^2 + 150^2) = sqrt(160000 + 22500) = sqrt(182500) ≈ 427.54 km
Now, let's find the direction of the resultant vector. We can use trigonometry to determine the angle between the resultant vector and the due south direction.
The tangent of this angle can be found using the formula:
tangent(angle) = magnitude_east / magnitude_south
In this case:
tangent(angle) = 150 km / 400 km = 0.375
Now, to find the angle, we need to take the arctangent of 0.375:
angle = arctan(0.375) ≈ 20.56 degrees
Therefore, the ship must travel approximately 20.56 degrees east of south to reach its destination.