Problem 51 - University Calculus: Early Transcendentals

Question

Location \(\quad\) 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

The tree is at (2.5, 4.33) km; the telephone pole is at (-4.57, -2.74) km.

Step 1: Define coordinates

We will identify the origin of our coordinate system at the bird's nest with coordinates \((0, 0)\). The positive \(x\)-axis points east, and the positive \(y\)-axis points north.

Step 2: Calculate tree location

The bird flies 5 km in the direction \(60^\circ\) north of east. Break this movement into x- and y-components using trigonometry:\[x_1 = 5\cos(60^\circ) = 5 \times \frac{1}{2} = 2.5 \, \mathrm{km}\]\[y_1 = 5\sin(60^\circ) = 5 \times \frac{\sqrt{3}}{2} \approx 4.33 \, \mathrm{km}\]Thus, the tree is located at the point \((2.5, 4.33)\).

Step 3: Calculate southeast direction angle

Due southeast is \(45^\circ\) south of east, which corresponds to an angle of \(135^\circ\) from the north. We need this angle to calculate movement in the new direction.

Step 4: Calculate total movement to telephone pole

The bird flies an additional 10 km in the southeast direction that makes an angle of \(-45^\circ\) (or equivalently \(135^\circ\) counterclockwise) from the east. Let's break this second movement into x and y components:\[x_2 = 10\cos(-45^\circ) = 10 \times \frac{\sqrt{2}}{2} \approx 7.07 \, \mathrm{km} \, \, (but negative direction)\]\[y_2 = 10\sin(-45^\circ) = -10 \times \frac{\sqrt{2}}{2} \approx -7.07 \, \mathrm{km}\]Since southeast corresponds to the negative x-direction, the bird actually moves \(-7.07\) km in the x-direction, and \(-7.07\) km in the y-direction.

Step 5: Determine final position

Starting at the tree \((2.5, 4.33)\), add the second movement:\[x = 2.5 - 7.07 \approx -4.57 \, \mathrm{km}\]\[y = 4.33 - 7.07 \approx -2.74 \, \mathrm{km}\]Thus, the telephone pole is located at approximately \((-4.57, -2.74)\).

Key Concepts

Coordinate System

In vector calculus, understanding coordinate systems is crucial for solving positional problems. A coordinate system helps in determining the exact location of a point in a plane, in this case using an xy-plane. The origin, denoted by coordinates \((0, 0)\), typically serves as a starting reference point.

• The positive x-axis points towards the east direction.
• The positive y-axis directs towards the north.

Using our problem, the bird's nest sits at the origin. As the bird moves, its positions on the plane change, represented by the respective coordinates from this reference point.

Trigonometry

Trigonometry is essential in determining movement and positioning in navigation. It helps in finding direction and dissection of movements into components in a plane using angles.When the bird flies "60° north of east," trigonometric functions help to break this movement into two parts:

• Cosine is used for horizontal (x-axis) calculations. For instance, \[x_1 = 5\cos(60°) = 2.5 \, \mathrm{km}\]
• Sine functions help with vertical (y-axis) calculations, such as \[y_1 = 5\sin(60°) \approx 4.33 \, \mathrm{km}\]

Thus, the understanding of angles in standard position and how they translate into coordinate movements is pivotal to solving many navigation problems. Again, when the bird flies "southeast," trigonometry allows us to understand this as a 45° angle adjustment from the primary axes. This decomposes the flight into further x and y movements, assisting in determining the exact locations on the map.

Vector Components

Vector components refer to breaking down a vector into perpendicular axes parts, generally along the x and y directions in a 2D coordinate system. Vectors are entities that possess both magnitude and direction:- **Magnitude**: tells how far the vector travels- **Direction**: tells where the vector is going, usually given in anglesWhen we analyze the bird's flight, vectors play a pivotal role.

• First, the bird's initial path of 5 km in the "60° north of east" direction is split into x-component (2.5 km) and y-component (4.33 km).
• For the second flight segment; the bird's 10 km journey is divided based on a 45° southeast direction angle (\[x_2 = -7.07\] and \[y_2 = -7.07\] moving negative relative to the axes).

These calculations allow for combining the vector components, illustrating how the bird moves in each direction independently. The resulting position comes from the summation of these vector components, effectively capturing the bird's entire journey on the coordinate plane.