Problem 45 - Precalculus Mathematics for Calculus

Question

Navigation A fisherman leaves his home port and heads in the direction \(\mathrm{N} 70^{\circ} \mathrm{W}\). He travels \(30 \mathrm{mi}\) and reaches Egg Island. The next day he sails \(\mathrm{N} 10^{\circ} \mathrm{E}\) for \(50 \mathrm{mi}\), reaching Forrest Island. (a) Find the distance between the fisherman's home port and Forrest Island. (b) Find the bearing from Forrest Island back to his home port.

Step-by-Step Solution

Answer

(a) About 77.42 miles; (b) Bearing: \( \mathrm{S} 88.8^{\circ} \mathrm{E} \).

Step 1: Understand the Problem

The problem involves navigation and requires calculating the distance and bearing between two locations. The fisherman travels in two legs: first towards \( \mathrm{N} 70^{\circ} \mathrm{W} \) for 30 miles and then \( \mathrm{N} 10^{\circ} \mathrm{E} \) for 50 miles. We need to find the direct distance and bearing from his original port to his final destination, Forrest Island.

Step 2: Establish the Coordinate System

Model the scenario using a coordinate plane where the home port is at the origin, \((0,0)\). His first leg of travel is towards \( \mathrm{N} 70^{\circ} \mathrm{W} \), meaning the angle is measured 70 degrees counter-clockwise from the negative x-axis (west to north). His second leg is towards \( \mathrm{N} 10^{\circ} \mathrm{E} \), an angle 10 degrees clockwise from the positive y-axis (north).

Step 3: Calculate Position After First Leg

Use trigonometry to find the coordinates after the first leg. \( x_1 = -30 \cos(70^{\circ}) \) and \( y_1 = 30 \sin(70^{\circ}) \). Compute these values to find the coordinates \((-10.26, 28.19)\) after 30 miles.

Step 4: Calculate Position After Second Leg

From the first checkpoint \((-10.26, 28.19)\), calculate his position after traveling 50 miles at \( \mathrm{N} 10^{\circ} \mathrm{E} \). The coordinates are \( x_2 = x_1 + 50 \sin(10^{\circ}) \) and \( y_2 = y_1 + 50 \cos(10^{\circ}) \). Resulting in coordinates \((-1.57, 77.41)\).

Step 5: Determine the Total Distance

The direct distance from the home port to Forrest Island is calculated using the distance formula \(d = \sqrt{(x_2 - 0)^2 + (y_2 - 0)^2} \), which simplifies to \(d = \sqrt{(-1.57)^2 + (77.41)^2}\). This results in a distance of approximately 77.42 miles.

Step 6: Find the Bearing from Forrest Island

Calculate the bearing back to home port from Forrest Island. Use the inverse tangent function \( \text{tan}^{-1}\left(\frac{77.41}{1.57}\right)\) and adjust to standard bearing format. Since it's initially measured from the east line moving clockwise, the final bearing is approximately \( \mathrm{S} 88.8^{\circ} \mathrm{E} \).

Key Concepts

Navigation Problems

Navigation problems often involve the use of maps and compass directions to determine the path and distance between two points. When navigating, understanding directions such as "North 70° West" is crucial. This direction means that you head 70 degrees towards the west from the north, like drawing an angle counter-clockwise from the north. Similarly, "North 10° East" involves turning clockwise just 10 degrees from the north. These directions are vital for determining the correct paths on the Earth’s surface. Navigation problems require precision, and this is where tools like compasses and maps come into play.
Whether it's navigating across open seas or plotting a course for an air journey, comprehending these bearings ensures you travel the intended path. Practical tasks such as fishing expeditions benefit greatly from this as they involve moving from point A to B using precise compass directions.

Bearing and Distance Calculation

Bearing and distance calculations form the core of navigation, especially in trigonometry. To solve problems involving distance and bearing, you can use a coordinate system based on a map.

• The bearing from a starting point gives the direction expressed in degrees.
• Calculating distance involves understanding how each leg of the journey contributes to the overall path.

To find a resultant distance, the journey can be split into legs, each with its respective bearings and distances, like vectors on a graph. For example, moving 30 miles on a heading of N 70° W changes your position using trigonometric functions: cosine for the east-west displacement and sine for the north-south displacement.
Subsequent legs are calculated in a similar manner. By plotting these calculated points on a coordinate system, you can determine the total direct distance using the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). This computation tells you how far you are from the starting point in a straight line.

Coordinate Geometry

Coordinate geometry offers a structured approach to solving navigation problems mathematically. By using an imaginary grid or coordinate plane, you can model real-world journeys efficiently. Starting with a point at the origin, every leg of your journey can be represented as a movement from this starting point. This approach allows for clear and accurate calculation of positions by using coordinates.

• Each point is represented by \(x, y\) coordinates that help trace the path on the plane.
• Movements are easily calculated using sine and cosine functions, considering the direction of movement as angles.

After plotting the entire journey, coordinate geometry aids in identifying finishing points and calculating distances. With Euler's distance formula, you can find the total linear distance between two points on the plane. For bearings, coordinate geometry allows you to determine angles using inverse trigonometric functions like the tangent inverse, providing a clear directional point from one location back to another. This interconnectedness makes coordinate geometry an invaluable tool in navigation scenarios.