A railroad track enters a field 480 feet north of the southwest corner and leaves 340 feet east of the southwest corner what angle does the track make with the south boundary?

you have a right triangle, with

vertical leg = 480
horizontal leg = 340

so, the angle θ from the horizontal base is

tanθ = 480/340

To find the angle made by the railroad track with the south boundary, we can break down the problem into steps:

Step 1: Visualize the situation
Draw a diagram representing the field and the railroad track. Place a coordinate system where the southwest corner has coordinates (0,0), and the north boundary is the positive y-axis, and the east boundary is the positive x-axis. Mark the entrance point of the railroad track (480 feet north of the southwest corner) and the exit point (340 feet east of the southwest corner).

Step 2: Determine the position of the exit point
Since the exit point of the railroad track is 340 feet east of the southwest corner, we can represent it with the coordinates (340, 0) on the coordinate system.

Step 3: Determine the position of the entrance point
Given that the entrance point of the railroad track is 480 feet north of the southwest corner, we can represent it with the coordinates (0, 480) on the coordinate system.

Step 4: Calculate the angle
The angle made by the railroad track with the south boundary can be found by calculating the angle between the line segment connecting the entrance point (0, 480) and the exit point (340, 0) and the positive x-axis.

To find this angle, we can use the inverse tangent function:

angle = arctan((change in y) / (change in x))

Using the coordinates, we have:

angle = arctan((480 - 0) / (340 - 0))

angle = arctan(480 / 340)

Using a calculator or a math software, we can find that:

angle ≈ 51.34 degrees

Therefore, the angle made by the railroad track with the south boundary is approximately 51.34 degrees.