Final answer:
Without a specific shape or dimensions to reference, locating the centroid of a plane area is impossible. For a rectangle, the centroid is at its center, but for complex shapes or density variations, integration is needed to determine the centroid by calculating area moments or mass moments divided by the total area or mass. The correct answer is option C
Step-by-step explanation:
The initial question doesn't provide adequate information or a figure for calculating the centroid of the specified plane area since we don't know the shape or dimensions beyond the single length of a = 45 in. To determine the centroid of a plane area, we typically divide the area into simpler shapes whose centroids we can locate easily, calculate the area moments about both x and y axes, and then find the centroid by dividing the sum of the moments by the total area.
For a rectangular area with uniform density, the centroid is simply at the center of the rectangle, which is at the coordinates (a/2, b/2) for a rectangle placed with one corner at the origin with sides of lengths a and b. When the density varies, such as a linear increase with xy as given by p(x, y) = p0xy, we integrate to find the position of the centroid by considering the density distribution.
In general, to find the centroid of a shape like a circular area that is concentric with the origin, it will be at the origin because the area is symmetric about both axes. For more complex geometries and density functions, integrating the moments and dividing by the total area or mass gives the centroid coordinates.