Solve each problem. See Examples 1 and 2. Distance between Two Ships A ship leaves its home port and sails on a bearing of S 61°50'. Another ship leaves the same port at the same time and sails on a bearing of N 28°10'E. If the first ship sails at 24.0 mph and the second sails at 28.0 mph, find the distance between the two ships after 4 hr.

Verified step by step guidance

1. Convert the bearings into standard angles measured counterclockwise from the positive x-axis (East). For the first ship with bearing S 61°50', this corresponds to an angle of 180° + 61°50' (since it's measured clockwise from North to South), so calculate the exact angle in degrees. For the second ship with bearing N 28°10' E, this corresponds to an angle of 90° - 28°10' (since it's measured clockwise from North to East), so calculate this angle as well.
2. Calculate the distance each ship has traveled after 4 hours by multiplying their speeds by time: Distance = Speed × Time. For the first ship, multiply 24.0 mph by 4 hours; for the second ship, multiply 28.0 mph by 4 hours.
3. Express the position of each ship in Cartesian coordinates (x, y) using their distances and angles. Use the formulas: \(x = r \times \cos(\theta)\) and \(y = r \times \sin(\theta)\), where \(r\) is the distance traveled and \(\theta\) is the angle in radians (convert degrees to radians).
4. Find the difference in the x-coordinates and y-coordinates between the two ships to get the components of the vector representing the distance between them: \(\Delta x = x_2 - x_1\) and \(\Delta y = y_2 - y_1\).
5. Use the Pythagorean theorem to find the distance between the two ships: \(\text{Distance} = \sqrt{(\Delta x)^2 + (\Delta y)^2}\). This will give the straight-line distance between the ships after 4 hours.

Key Concepts

Bearing and Direction in Navigation

Bearing is a way to describe direction using angles measured clockwise from the north or south line. Bearings like S 61°50' mean starting from south and rotating 61°50' towards the east. Understanding how to convert these bearings into standard angles or vectors is essential for plotting the ships' courses accurately.

Vector Representation of Displacement

Each ship's movement can be represented as a vector based on its speed, time traveled, and direction (bearing). Calculating the displacement vectors allows us to find the ships' positions relative to the port after 4 hours, which is necessary to determine the distance between them.

Law of Cosines for Distance Calculation

When two vectors form an angle, the distance between their endpoints can be found using the Law of Cosines. This law relates the lengths of two sides of a triangle and the included angle to find the third side, which in this problem is the distance between the two ships after traveling.