91Ó°ÊÓ

The Cartesian Coordinate System is a way to represent points in space using a pair of numerical coordinates. These coordinates are based on two fixed perpendicular lines, usually called axes. The horizontal axis is known as the x-axis, and the vertical axis as the y-axis.
In the context of the exercise, the position of the two towns is plotted on the Cartesian plane. We assume one town, Town A, is at the origin, (0, 0). Therefore, Town B, which is 72 km west and 35 km south of Town A, is placed at the coordinate (-72, -35).

• The negative sign indicates direction: west (negative x-direction) and south (negative y-direction).
• This coordinate system helps in visualizing distances and directions in a two-dimensional plane.

Understanding this setup makes it easier to solve problems involving distance and direction.

To calculate the shortest distance between two points on a plane, the Pythagorean theorem comes in handy. This theorem applies because the two points and the differences in their x and y coordinates form a right triangle.
The formula according to the Pythagorean theorem is \[ c = \sqrt{a^2 + b^2} \]
where \(a\) and \(b\) are the lengths of the two sides of the triangle, and \(c\) is the hypotenuse, representing the shortest distance.

• The length \(a\) is the difference along the x-axis (72 km).
• The length \(b\) is the difference along the y-axis (35 km).

Substituting these values into the formula, we get:\[ c = \sqrt{72^2 + 35^2} = \sqrt{5184 + 1225} = \sqrt{6409} \approx 80.07 \text{ km}\]
This result represents the shortest possible highway distance between the two towns.

Trigonometry, the study of triangles, specifically right triangles, aids in finding more than just distances; it helps determine angles.
When you have a right triangle, as created by the two points on the Cartesian plane, trigonometric functions relate the angles of the triangle to the lengths of its sides.
In this case, we are interested in finding the angle between the shortest path (hypotenuse) and the horizontal x-axis.

• The tangent function, \( \tan(\theta) \), relates the angle \(\theta\) to the opposite side (difference in y coordinates) and the adjacent side (difference in x coordinates).

The formula used here is:\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{35}{72} \]
From this, we can use the inverse tangent function to find \(\theta\). Understanding trigonometry is crucial for solving real-world problems involving direction and angles.

Once we have the tangent of the angle, the next step is to calculate the angle itself. This angle is measured from the west direction.
To find this angle, we use the arctangent function, denoted by \( \arctan \). The arctangent function will give us the measure of the angle whose tangent is a given number.
The calculation is:\[ \theta = \arctan\left(\frac{35}{72}\right) \approx 26.57^\circ \]
This result tells us that the highway makes an angle of approximately 26.57° south of due west.

• An understanding of angles is essential in navigation and construction, as it dictates the precise direction in which something like a highway should be built.

Measuring angles correctly ensures that the constructed path is both optimal and accurate.