Skip to main content
Ch. 2 - Acute Angles and Right Triangles
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 2 - Acute Angles and Right TrianglesProblem 2.5.25
Chapter 3, Problem 2.5.25

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.
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.
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).
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 \(\Delta y = y_2 - y_1\).
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
05:13
Finding Direction of a Vector

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.
Recommended video:
03:48
Introduction to Vectors

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.
Recommended video:
4:35
Intro to Law of Cosines
Related Practice
Textbook Question

Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.

1/csc(90°-51°)

2
views
Textbook Question

Concept Check The two methods of expressing bearing can be interpreted using a rectangular coordinate system. Suppose that an observer for a radar station is located at the origin of a coordinate system. Find the bearing of an airplane located at each point. Express the bearing using both methods. (-3, -3)

Textbook Question

Use a calculator to approximate the value of each expression. Give answers to six decimal places. In Exercises 21–28, simplify the expression before using the calculator. See Example 1.

cos 41° 24'

1
views
Textbook Question

Use a calculator to determine whether each statement is true or false. A true statement may lead to results that differ in the last decimal place due to rounding error. cos(30° + 20°) = cos 30° + cos 20°

1
views
Textbook Question

Solve each problem. See Examples 3 and 4. The figure to the right indicates that the equation of a line passing through the point (a, 0) and making an angle θ with the x-axis is y = (tan θ) (x - a). Find an equation of the line passing through the point (5, 0) that makes an angle of 15° with the x-axis.

5
views
Textbook Question

Match each trigonometric function in Column I with its value in Column II. Choices may be used once, more than once, or not at all.

sin 30°

5
views