Problem 47 - Thomas Calculus

Question

Location \(A\) bird flies from its nest 5 \(\mathrm{km}\) in the direction \(60^{\circ}\) north of east, where it stops to rest on a tree. It then flies 10 \(\mathrm{km}\) in the direction due southeast and lands atop a telephone pole. Place an \(x y\) -coordinate system so that the origin is the bird's nest, the \(x\) -axis points east, and the \(y\) -axis points north. a. At what point is the tree located? b. At what point is the telephone pole?

Step-by-Step Solution

Answer

Tree: (2.5, 4.33) km; Telephone Pole: (-4.57, -2.74) km.

Step 1: Analyze Bird's First Flight

The bird flies 5 km at an angle of \(60^{\circ}\) north of east. This means it forms a right triangle where the angle from the x-axis (east) is \(60^{\circ}\). The x-component (east) and y-component (north) can be calculated using cosine and sine respectively.

Step 2: Calculate Coordinates of the Tree

To find the x-coordinate, use \(x_1 = 5 \cdot \cos(60^{\circ})\). For the y-coordinate, use \(y_1 = 5 \cdot \sin(60^{\circ})\). By calculating, you get \(x_1 = 5 \cdot 0.5 = 2.5\) km and \(y_1 = 5 \cdot \frac{\sqrt{3}}{2} \approx 4.33\) km. The coordinates of the tree are \((2.5, 4.33)\).

Step 3: Analyze Bird's Second Flight

The bird then flies 10 km in the direction of southeast. Southeast corresponds to \(135^{\circ}\) from the east (x-axis), forming an angle of \(45^{\circ}\) from the south in the clockwise direction from north.

Step 4: Calculate Coordinates of the Telephone Pole

Convert the southeast direction into components. The x-component is negative as the flight moves in the southwest direction: \(x_2 = 10 \cdot \cos(135^{\circ})\). The y-component is also negative as it moves downwards: \(y_2 = 10 \cdot \sin(135^{\circ})\). This yields \(x_2 = 10 \cdot (-\frac{\sqrt{2}}{2}) \approx -7.07\) km and \(y_2 = 10 \cdot (-\frac{\sqrt{2}}{2}) \approx -7.07\) km. The final position of the telephone pole from the origin using this direction from the tree point is \(x=2.5-7.07\approx -4.57\) and \(y= 4.33-7.07 \approx -2.74\), thus the coordinates of the telephone pole are \((-4.57, -2.74)\).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies triangles, particularly the relationships between their angles and sides. In this exercise, trigonometry plays a key role in determining the bird's position as it flies in specific directions.

• To calculate positions using angles and distances, we use trigonometric functions like sine and cosine.
• Sine (\( ext{sin}\)) helps find the length of the side opposite an angle when the hypotenuse is known.
• Cosine (\( ext{cos}\)) helps find the length of the adjacent side under the same conditions.

For the bird's first flight, which is 5 km at an angle of \(60^{\circ}\) north of east, we need these functions:
\[ x_1 = 5 \cdot \cos(60^{\circ}) \] \[ y_1 = 5 \cdot \sin(60^{\circ}) \] These calculations give us the x and y-coordinates of the tree's location. Trigonometry provides a systematic way to quantify direction and distance in coordinate geometry.

Vector Components

Vectors are mathematical objects used to represent quantities that have both magnitude and direction, like the bird's flights in this scenario. When breaking down the bird's movement, we use vector components to estimate its location step-by-step.
Each flight creates a vector with components along the x and y axes, representing the east-west and north-south directions, respectively.

• The x-component indicates movement parallel to the x-axis (east).
• The y-component reflects movement parallel to the y-axis (north).

For the 5 km flight at \(60^{\circ}\), the components are calculated with:
\[ x_1 = 5 \cdot \cos(60^{\circ}) \] \[ y_1 = 5 \cdot \sin(60^{\circ}) \] For the 10 km flight southeast, recognize southeast as \(135^{\circ}\) from east, giving:
\[ x_2 = 10 \cdot \cos(135^{\circ}) \] \[ y_2 = 10 \cdot \sin(135^{\circ}) \] Remember, if the direction is southwest or downward, the vector components could be negative, indicating movement in the opposite direction.

Right Triangle

Right triangles are central to coordinate geometry problems involving angles because they are the simplest form of triangle and have one angle precisely \(90^{\circ}\). This property vastly simplifies calculations involving distances and angles in the Cartesian plane.
In such triangles, once we know the angle and at least one side, we can use trigonometric functions to find missing parts.

• The side opposite an angle can be determined using sine (\( ext{sin}\)).
• The adjacent side can be calculated using cosine (\( ext{cos}\)).

In this exercise, the right triangle helps analyze both flights. For the first flight, drawn as a right triangle, \(5 \, \text{km}\) is the hypotenuse, while the x and y components represent the other two sides, calculated as:
\[ x_1 = 5 \cdot \cos(60^{\circ}) \] \[ y_1 = 5 \cdot \sin(60^{\circ}) \] For the second flight southeast, remember it complements into another right triangle, even while aiming at non-cardinal directions. Calculations critically hinge on this geometric concept.