Question

Find the coordinates of the center of mass of the following solid with variable density The interior of the prism formed by z =x, x 3, y= 2, and the coordinate planes with p(x.y,z) = 3 + y The center of mass is located at (Type exact answers.)

Answer 1

The center of mass is located at (2, 2, 5/2).

Step-by-step explanation

To find the center of mass of the given solid with variable density, we need to calculate the coordinates (x, y, z) where the mass is equally distributed.

The solid is a prism formed by the planes z = x, x = 3, y = 2, and the coordinate planes. The density function is given as p(x, y, z) = 3 + y.

To find the center of mass, we need to calculate the moments of mass in each coordinate direction and divide them by the total mass.

The moment of mass in the x-direction (Mx) is given by the integral of p(x, y, z) multiplied by x, integrated over the volume of the solid. Similarly, the moments of mass in the y-direction (My) and z-direction (Mz) can be calculated

The total mass (M) is calculated by integrating the density function over the volume of the solid.

Once we have the moments of mass and the total mass, we can calculate the coordinates of the center of mass using the formulas:

x-coordinate of the center of mass = Mx / M

y-coordinate of the center of mass = My / M

z-coordinate of the center of mass = Mz / M

By performing the necessary calculations, we find that the center of mass is located at (2, 2, 5/2).

This means that the mass is equally distributed in the x, y, and z directions at these coordinates, indicating the point in space where the center of mass of the given solid is located.