Final answer:
The center of mass of a thin plate with constant density covering the region given by the question is located at the centroid of the triangular area bounded by y=5-2x, the x-axis, and the y-axis. The centroid coordinates are calculated as (0.833, 1.667).
Step-by-step explanation:
Finding the Center of Mass of a Thin Plate
To find the center of mass of a thin plate with constant density within the specified boundaries, we need to set up the integral for the center of mass using the known geometric properties. The region is bounded by the line y = 5 - 2x, and the positive x and y axes. The boundaries define a triangular area within the first quadrant. Since the density is constant, the center of mass will be at the centroid of this triangle.
To calculate the coordinates of the centroid (center of mass), we use the formulas for finding the centroid of a triangle. The x-coordinate is the average of the x-coordinates of the vertices, and the y-coordinate is the average of the y-coordinates of the vertices. In this case, the vertices of the triangle are located at (0,0), (0,5), and (2.5,0). Thus, the x-coordinate of the center of mass is x = (0 + 0 + 2.5) / 3 = 0.833, and the y-coordinate is y = (0 + 5 + 0) / 3 = 1.667.
Therefore, the center of mass is located at (0.833, 1.667). This assumes that the plate is uniformly dense and of negligible thickness, which simplifies the calculation significantly as we do not need to take the z-coordinate into account.