Question

You are camping with 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 m from yours, in the direction 23.0\(^{\circ}\) south of east. Karl's tent is 32.0 m from yours, in the direction 37.0\(^{\circ}\) north of east. What is the distance between Karl's tent and Joe's tent?

Short answer

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

Step-by-step solution

Understand the Coordinates

Imagine your position as the origin (0, 0) on a coordinate plane. Joe's tent is 21.0 meters away, 23.0 degrees south of east. Karl's tent is 32.0 meters away, 37.0 degrees north of east.

Convert to Cartesian Coordinates

First, we convert these polar coordinates into rectangular coordinates. For Joe's tent, the position is given by:\[ x_1 = 21.0 \cos(23.0^{\circ}) \]\[ y_1 = -21.0 \sin(23.0^{\circ}) \]For Karl's tent, the position is given by:\[ x_2 = 32.0 \cos(37.0^{\circ}) \]\[ y_2 = 32.0 \sin(37.0^{\circ}) \]

Calculate Coordinates

Calculate the coordinates for Joe's tent:\[ x_1 = 21.0 \times 0.9205 \approx 19.33 \, \text{m} \]\[ y_1 = -21.0 \times 0.3907 \approx -8.21 \, \text{m} \]Calculate the coordinates for Karl's tent:\[ x_2 = 32.0 \times 0.7986 \approx 25.55 \, \text{m} \]\[ y_2 = 32.0 \times 0.6018 \approx 19.26 \, \text{m} \]

Determine Distance Formula

Use the distance formula to determine the distance between Karl's tent and Joe's tent:The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 } \]

Calculate the Distance

Substitute the coordinates into the distance formula to determine the final distance:\[ d = \sqrt{(25.55 - 19.33)^2 + (19.26 - (-8.21))^2 } \]\[ d = \sqrt{(6.22)^2 + (27.47)^2 } \]\[ d = \sqrt{38.71 + 754.61} \]\[ d \approx \sqrt{793.32} \approx 28.15 \, \text{m} \]

Conclusion

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

Key concepts

Cartesian Coordinates

Imagine a grid made of horizontal (x-axis) and vertical (y-axis) lines intersecting at right angles. This grid forms a plane, commonly referred to as the Cartesian coordinate system. Using this system, any point on the plane can be described with an

• x-coordinate, which tells you how far left or right the point is from the origin, and
• a y-coordinate, which shows how far up or down the point is from the origin.

For example, in the tent camping problem, the origin is where your tent is pitched, designated as (0, 0) on the coordinate plane. Joe's and Karl's tents have coordinates derived from their distances and directions. This coordinate system makes it simpler to calculate distances between two points, which is very handy in geometry and everyday problems! When converting from directions like south of east or north of east to Cartesian coordinates, trigonometric functions, such as sine and cosine, help find the specific x and y values.

Distance Formula

The distance formula is a mathematical way to determine the distance between two points in a coordinate plane. It's based on the Pythagorean theorem and uses the x and y coordinates of the points. The formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Here,

• \(x_1\) and \(x_2\) are the x-coordinates of the two points,
• \(y_1\) and \(y_2\) are the y-coordinates.

To get the straight-line distance between Joe's and Karl’s tents, you apply this formula using their calculated coordinates. By substituting these values,

• you find the differences in x and y,
• square these differences, add them, and lastly, take the square root of the sum.

This gives the exact distance between the two tents, which, in the example from the exercise, results in approximately 28.15 meters. Understanding how each part of the formula contributes to the overall calculation is crucial for grasping the concept of distances in coordinates.

Polar Coordinates

Unlike Cartesian coordinates which use lengths along axes to describe a location, polar coordinates utilize a radius and angle.

• The radius indicates how far away from a fixed point (the pole) a point is,
• and the angle specifies the direction relative to a reference direction, often the positive x-axis.

In scenarios such as the tent example, angles like 23 degrees south of east or 37 degrees north of east are used. To work with these in a Cartesian coordinate system, we need to convert them from polar to rectangular forms.

• The x-coordinate can be found using \( r \cos(\theta) \)
• and the y-coordinate via \( r \sin(\theta) \), where \( r \) is the radius, and \( \theta \) the angle.

This conversion plays a critical role in transforming directions and distances into a format that's easier to handle mathematically, especially when precise calculations or comparisons are necessary.