Find the center of mass for the region bound by y= -2x+6, x=0, and y=0
To find the center of mass for the region bounded by the equations y = -2x + 6, x = 0, and y = 0, we need to determine the x-coordinate and y-coordinate of the center of mass.
First, let's visualize the region by graphing the given equations on a coordinate plane.
The equation y = -2x + 6 represents a straight line with a y-intercept of 6 and a slope of -2. The line intersects the y-axis at point (0, 6) and the x-axis at point (3, 0).
Considering the region is bounded by x = 0 and y = 0, we have a triangle. The vertices of the triangle are (0, 0), (0, 6), and (3, 0).
To find the x-coordinate of the center of mass, we need to calculate the average of the x-coordinates of the three vertices. In this case, since one of the vertices is at (0, 0), the average of the x-coordinates is:
x̄ = (0 + 0 + 3) / 3 = 3 / 3 = 1
So, the x-coordinate of the center of mass is 1.
Next, to find the y-coordinate of the center of mass, we need to calculate the average of the y-coordinates of the three vertices. In this case, the y-coordinates are 0, 6, and 0, so the average is:
ȳ = (0 + 6 + 0) / 3 = 6 / 3 = 2
Therefore, the y-coordinate of the center of mass is 2.
In conclusion, the center of mass for the region bounded by y = -2x + 6, x = 0, and y = 0 is located at the point (1, 2).