Problem 45 - A Graphical Approach to Precalculus with Limits

Question

A ship leaves port and sails on a bearing of \(\mathrm{N} 28^{\circ} 10^{\prime} \mathrm{E}\). Another ship leaves the same port at the same time and sails on a bearing of \(\mathrm{S} 61^{\circ} 50^{\prime} \mathrm{E} .\) If the first ship sails at \(24.0 \mathrm{mph}\) and the second sails at \(28.0 \mathrm{mph}\), find the distance \(x\) between the two ships after 4 hours.

Step-by-Step Solution

Answer

The distance between the two ships after 4 hours is 147.5 miles.

Step 1: Determine the distance each ship has traveled

First, we calculate the distance traveled by each ship after 4 hours. The first ship travels at 24 mph, so in 4 hours it travels \(24 \times 4 = 96 \text{ miles}\). The second ship travels at 28 mph, thus covering \(28 \times 4 = 112 \text{ miles}\).

Step 2: Convert bearings to angles

The first ship's bearing is \(28^{\circ} 10'\) east of north, which translates directly to an angle from north. The second ship's bearing is \(61^{\circ} 50'\) east of south, which is equivalent to \(180^{\circ} - 61^{\circ} 50' = 118^{\circ} 10'{\text{ from north}}.\)

Step 3: Calculate the angle between the two ship paths

To find the angle between the paths the ships are taking, find the difference between the angles based on a north reference: \(118^{\circ} 10' - 28^{\circ} 10' = 90^{\circ}.\) Hence, the paths are perpendicular (90 degrees between them).

Step 4: Use the Pythagorean theorem to find the distance between the ships

Since the paths are at a right angle to each other, form a right triangle with the distances as the two legs: \(a = 96 \text{ miles}\) and \(b = 112 \text{ miles}.\) Use the Pythagorean Theorem: \[ c = \sqrt{96^2 + 112^2} = \sqrt{9216 + 12544} = \sqrt{21760} = 147.5 \text{ miles}.\]

Step 5: Conclusion

The distance \(x\) between the two ships after 4 hours is calculated as \(147.5\) miles using the derived right triangle relationship from the bearings.

Key Concepts

Bearing angles

Bearing angles are used to describe the direction in which an object is moving. In navigation, they are crucial for plotting courses and making sure vessels reach their intended destinations. A bearing is typically measured in degrees from a point of reference, usually North or South.

• N 28° 10' E: This means the direction is 28 degrees East of North.
• S 61° 50' E: This indicates the direction is 61 degrees East of South.

Bearings follow a clockwise motion from North unless specified otherwise. To effectively understand and manipulate bearings in problems, convert them to angles measured from a single reference line, such as North, as demonstrated in this exercise.
Understanding how to break down and translate these bearings is a key skill for solving navigational and trigonometry problems.

Pythagorean theorem

The Pythagorean Theorem is a fundamental principle in geometry, specifically applicable to right triangles. It relates the lengths of the three sides: the two legs and the hypotenuse. The theorem is expressed by the equation:\[c^2 = a^2 + b^2\]where \(a\) and \(b\) are the triangle's legs, and \(c\) is the hypotenuse.
This exercise uses the Pythagorean Theorem to find the straight-line distance between two ships. Since the paths form a right triangle, these two paths are the legs, \(a = 96\) miles and \(b = 112\) miles. The theorem allows calculating the hypotenuse, represented by:\[c = \sqrt{96^2 + 112^2}\]
Thus, solving this provides the distance between the two points, or ships in this case.

Distance calculation

Calculating distances is a common application of trigonometry and geometry. When a problem involves straight-line distances between points, identifying right triangles is often a key strategy to solve it.

• Determine path lengths: Multiply speed by time for each path.
• Identify geometric relationships: Recognize perpendicular paths confirming a right-angle triangle.
• Plug into the Pythagorean Theorem: This calculates the direct distance between two points.

In the given problem, the task involved calculating the distances each ship traveled (96 miles and 112 miles) and using these lengths to find the direct distance between them after 4 hours. Such straightforward step-by-step calculations facilitate clearer understanding and application of trigonometric principles.

Right triangles

Right triangles are a type of triangle with one 90-degree angle. They are critical to many real-world geometry problems because they simplify relationships between sides and angles. An essential property of right triangles is the Pythagorean Theorem, allowing calculations involving lengths.
In this exercise, the right triangle forms as ships sail at perpendicular bearings. By interpreting these paths as legs of a triangle, we can use the theorem to solve for the hypotenuse, representing the distance sought.
When approaching right triangle problems:

• Identify or determine the right angle.
• Use known legs to solve for unknown lengths, often the hypotenuse when dealing with distance.
• Interpret real-world directions and angles accurately, considering the specific geometry imposed.

This approach not only aids in solving navigation problems but also builds a foundational understanding applicable to broader trigonometric problems.