Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Jogger's course A jogger runs at a constant speed of one mile every 8 minutes in the direction \(\mathrm{S} 40^{\circ} \mathrm{E}\) for 20 minutes and then in the direction \(\mathrm{N} 20^{\circ} \mathrm{E}\) for the next 16 minutes. Approximate, to the nearest tenth of a mile, the straightline distance from the endpoint to the starting point of the jogger's course.

Step-by-Step Solution

Answer

The straight-line distance is approximately 1.4 miles.

Step 1: Calculate the Distance for Each Segment

The jogger runs at a speed of one mile every 8 minutes. First, calculate the distance covered in each segment. For the first segment, the jogger runs for 20 minutes. The distance covered is \( \frac{20}{8} = 2.5 \) miles. For the second segment, the jogger runs for 16 minutes. The distance covered is \( \frac{16}{8} = 2 \) miles.

Step 2: Convert Directions into Vectors

Convert the jogger's directional movements into vectors using trigonometry. The first segment's direction is \( \mathrm{S} 40^{\circ} \mathrm{E} \), which can be represented as a vector: \((-2.5 \sin(40^{\circ}), -2.5 \cos(40^{\circ}))\). The second segment's direction is \( \mathrm{N} 20^{\circ} \mathrm{E} \), represented as \((2 \sin(20^{\circ}), 2 \cos(20^{\circ}))\). Note the signs based on the compass direction system.

Step 3: Calculate Resultant Endpoint Vector

Sum the vectors from each segment to find the resultant vector. The x-component is \( -2.5 \sin(40^{\circ}) + 2 \sin(20^{\circ}) \). The y-component is \( -2.5 \cos(40^{\circ}) + 2 \cos(20^{\circ}) \).

Step 4: Compute the Straight-Line Distance

The resultant vector gives the position from the starting point. Use the Pythagorean theorem to find the straight-line distance: \( \sqrt{(-2.5 \sin(40^{\circ}) + 2 \sin(20^{\circ}))^2 + (-2.5 \cos(40^{\circ}) + 2 \cos(20^{\circ}))^2} \). Calculate this expression to find the distance.

Key Concepts

Vector Addition

In trigonometry, vector addition is a crucial concept when dealing with forces or movements that have both a magnitude and a direction. A jogger's movement can be represented as vectors, where each vector is defined by the distance covered in a specific direction.

Vectors are measured in terms of their horizontal (x) and vertical (y) components, which can be calculated using sine and cosine functions of the given angles. For example:

• The vector for the jogger's first segment in the direction of \( S 40^{\circ} E \) has the components:
• \((-2.5 \sin(40^{\circ}), -2.5 \cos(40^{\circ}))\).

The negative signs indicate a direction different from the positive x or y axis.

Similarly, the second segment vector from direction \( N 20^{\circ} E \) is calculated as:

• \((2 \sin(20^{\circ}), 2 \cos(20^{\circ}))\).

Vector addition requires summing these components separately to determine the resultant vector. This gives a complete representation of the jogger's final position relative to the start.

By understanding vector addition, you can predict overall displacements and evaluate total paths traveled.

Compass Directions

Understanding compass directions is fundamental when navigating courses or paths like the one described in this exercise. Compass directions use the cardinal points: North, South, East, and West, to define angles between them.
The directions \( S 40^{\circ} E \) and \( N 20^{\circ} E \) indicate the precise orientation of the jogger's path on the compass.

• \( S 40^{\circ} E \) means starting from the south and moving 40 degrees towards the east.
• \( N 20^{\circ} E \) indicates starting from the north and heading 20 degrees towards the east.

By converting these compass directions into vectors with sine and cosine, you can map the path geometrically.

Compass direction facilitates understanding of travel routes in terms of angles and can be essential in fields like navigation, aviation, and geography. These angles dictate how vectors are formed and their components affected in any calculation or analysis involving them.

Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

For the jogger's path, once we know the components of the resultant vector, we apply the Pythagorean theorem to find the straight-line distance from start to finish.

The resultant vector components are summed:

• The x-component: \(-2.5 \sin(40^{\circ}) + 2 \sin(20^{\circ})\)
• The y-component: \(-2.5 \cos(40^{\circ}) + 2 \cos(20^{\circ})\)

Using the Pythagorean theorem, the straight-line distance is calculated as:
\[ \sqrt{(-2.5 \sin(40^{\circ}) + 2 \sin(20^{\circ}))^2 + (-2.5 \cos(40^{\circ}) + 2 \cos(20^{\circ}))^2} \]

This calculation results in the jogger's direct distance from the original point, representing the shortest route in straight line terms. Understanding and applying the Pythagorean theorem helps in finding distances and is widely applicable across physics, mathematics, engineering, and even computer science.