Solve each problem. See Examples 1 and 2. Distance between Two Ships Two ships leave a port at the same time. The first ship sails on a bearing of 52° at 17 knots and the second on a bearing of 322° at 22 knots. How far apart are they after 2.5 hr?

Verified step by step guidance

1. Identify the distances each ship travels by multiplying their speeds by the time traveled. For the first ship: \(\text{distance}_1 = 17 \times 2.5\), and for the second ship: \(\text{distance}_2 = 22 \times 2.5\).
2. Convert the bearings into angles relative to a common reference axis, typically the positive x-axis (east). Bearing is measured clockwise from north, so convert each bearing to standard position angles: \(\theta_1 = 90^\circ - 52^\circ\) and \(\theta_2 = 90^\circ - 322^\circ\) (adjusting angles to be between 0° and 360° as needed).
3. Represent the position of each ship as coordinates using their distances and angles: \(\text{Ship 1 coordinates} = (d_1 \cos(\theta_1), d_1 \sin(\theta_1))\) and \(\text{Ship 2 coordinates} = (d_2 \cos(\theta_2), d_2 \sin(\theta_2))\).
4. Calculate the difference in x-coordinates and y-coordinates between the two ships: \(\Delta x = x_2 - x_1\) and \(\Delta y = y_2 - y_1\).
5. Use the distance formula to find how far apart the ships are: \(\text{distance} = \sqrt{(\Delta x)^2 + (\Delta y)^2}\).

Key Concepts

Bearing and Direction in Navigation

Bearing is the direction or path along which something moves, measured in degrees clockwise from the north. Understanding bearings like 52° and 322° helps determine the exact direction each ship travels relative to the port, which is essential for plotting their positions on a coordinate plane.

Distance Calculation Using Speed and Time

Distance traveled is calculated by multiplying speed by time. Here, each ship's speed in knots (nautical miles per hour) and the travel time of 2.5 hours allow us to find how far each ship has moved from the port, which is necessary to determine their relative positions.

Law of Cosines for Distance Between Two Points

The Law of Cosines relates the lengths of sides of a triangle to the cosine of one angle. By modeling the ships' paths as two sides of a triangle with a known angle between their bearings, this law helps calculate the direct distance between the two ships after 2.5 hours.