A bike trail starts at (4.2, -30.5) and ends at (4.2, 9.7) on a map. Each unit represents 1 kilometer. What is the distance from the start of the trail to the end?
The bike trail is defined by its start point \(S = (4.2, -30.5)\) and its end point \(E = (4.2, 9.7)\) on a coordinate plane. Each unit on this coordinate plane corresponds to \(1\) kilometer. Since the x-coordinates of both points are the same (\(x_1 = x_2 = 4.2\)), the trail forms a perfectly vertical line segment on the map. Therefore, the total distance can be found by calculating the vertical distance between the two points.
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) with the same x-coordinate is given by the absolute difference of their y-coordinates:
\[d = |y_2 - y_1|\]
Substituting the given y-coordinates \(y_1 = -30.5\) and \(y_2 = 9.7\):
\[d = |9.7 - (-30.5)|\]
\[d = |9.7 + 30.5|\]
\[d = 40.2\]
Since each unit represents \(1\) kilometer, the distance of the bike trail is \(40.2\) kilometers.
Imagine you are planning a bike trip on a straight path marked on a grid map. If your starting position is at an elevation coordinate of \(-30.5\) and your destination is at an elevation coordinate of \(9.7\), calculating the vertical distance helps you estimate the length of your journey. Using the absolute difference formula \(\Delta y = |y_2 - y_1|\), you can quickly determine that the travel distance is \(40.2\) kilometers, allowing you to prepare your gear and water supply accordingly.