Problem 84 - University Physics with Modern Physics

Question

You are camping with two friends, Joe and Karl. Since all three of you like your privacy, you don't pitch your tents close together. Joe's tent is 21.0 \(\mathrm{m}\) from yours, in the direction \(23.0^{\circ}\) south of east. Karl's tent is 32.0 \(\mathrm{m}\) from yours, in the direction \(37.0^{\circ}\) north of east. What is the distance between Karl's tent and Joe's tent?

Step-by-Step Solution

Answer

The distance between Karl's tent and Joe's tent is approximately 28.15 meters.

Step 1: Understand the Problem

We need to find the distance between Karl's tent and Joe's tent, knowing their distances and directions relative to our tent.

Step 2: Represent the Problem with a Diagram

Create a coordinate system with our tent at the origin. Represent Joe's tent at the coordinates derived from his distance and angle from the origin, and do the same for Karl's tent.

Step 3: Convert Directional Distances to Coordinates

The x-coordinate for Joe's tent is \(21.0 \cos (23.0^{\circ})\) and the y-coordinate is \(-21.0 \sin (23.0^{\circ})\). For Karl's tent, the x-coordinate is \(32.0 \cos (37.0^{\circ})\) and the y-coordinate is \(32.0 \sin (37.0^{\circ})\).

Step 4: Compute the Coordinates

Calculate Joe's tent's coordinates: \((21.0 \cos (23.0^{\circ}), -21.0 \sin (23.0^{\circ})) \approx (19.35, -8.19)\). Calculate Karl's tent's coordinates: \((32.0 \cos (37.0^{\circ}), 32.0 \sin (37.0^{\circ})) \approx (25.56, 19.27)\).

Step 5: Use the Distance Formula

Apply the distance formula to find the distance between Joe's and Karl's tents: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), where \((x_1, y_1)\) are Joe's coordinates and \((x_2, y_2)\) are Karl's coordinates.

Step 6: Substitute and Calculate

Substitute the values into the distance formula: \(\sqrt{(25.56 - 19.35)^2 + (19.27 - (-8.19))^2} = \sqrt{(6.21)^2 + (27.46)^2}\). Calculate the result: \(\sqrt{38.56 + 754.44}\).

Step 7: Final Calculation

Simplify to get \(\sqrt{793.00} = 28.15.\) So, the distance between Karl's tent and Joe's tent is approximately \(28.15\) meters.

Key Concepts

Distance Formula

The distance formula is a vital tool in mathematics, particularly useful in finding the distance between two points in a coordinate system. Imagine having two points,

• Point A with coordinates \((x_1, y_1)\)
• Point B with coordinates \((x_2, y_2)\)

The distance \(d\) between these two points can be found using the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.\]This equation is derived from the Pythagorean theorem, which connects the horizontal distance \((x_2 - x_1)\) and vertical distance \((y_2 - y_1)\) between the two points to their total distance. The reason it involves squaring and then taking the square root of these differences is to ensure the distance is a positive value, as distance cannot be negative. In exercises like this, where exact coordinates are used, this formula simplifies the process by eliminating the need for a physical measuring tape!

Coordinate System

A coordinate system is like a map that helps us describe the exact position of points in space. Two-dimensional coordinates use X (horizontal) and Y (vertical) axes, and each axis is marked with numbers to locate points easily. By placing Joe's and Karl's tents on this map, we seamlessly translate their distances and directions into comprehensible data that we can compute. It's similar to plotting cities on a map based on their latitude and longitude. Once we have a coordinate system, it's much easier to visualize problems and apply mathematical tools, like the distance formula, for problem-solving. In our exercise, using a coordinate system helps comprehend how Joe and Karl's tents are positioned relative to our starting point, simplifying the task of finding the distance between them.

Vector Components

Vectors are quantities that have both magnitude and direction, making them incredibly useful in physics and geometry. When breaking down vectors into components, we split them into horizontal (x-axis) and vertical (y-axis) parts. In this exercise, we convert the distances and directions from Joe and Karl's tents into vector components. Joe's vector components are calculated as follows:

• x-component: \(21.0 \cos(23.0^\circ)\)
• y-component: \(-21.0 \sin(23.0^\circ)\)

Karl's vector components are calculated as:

• x-component: \(32.0 \cos(37.0^\circ)\)
• y-component: \(32.0 \sin(37.0^\circ)\)

These calculations transform angle and distance data into coordinates, allowing us to easily find the tents' exact positions on the coordinate map. Understanding vector components is crucial as it enables solving diverse problems as well as simplifying seemingly complex tasks.

Angle Conversion

When working with angles in trigonometry, it's often crucial to convert between degrees and radians for accurate calculations. However, in our original problem, directions were already provided in degrees, making our task easier. Nonetheless, understanding angle conversion is vital in many mathematical applications. An angle given in degrees can be converted into radians using:\[\text{radians} = \text{degrees} \times \frac{\pi}{180^\circ}\]Conversely, an angle in radians can be converted to degrees by:\[\text{degrees} = \text{radians} \times \frac{180^\circ}{\pi}\]In our problem, the angles helped convert Joe and Karl's locations into coordinates effortlessly. This angle use allows us to locate points precisely, reflecting direction in addition to magnitude. Understanding how to work with angles will aid you with more complex trigonometric problems in the future.