In order to determine a submarine's buoyancy characteristics, designers
must accurately determine the centroid, center of mass, and center of
gravity for various parts of the sub.

What is known:

The radius of the nose cone is given by
r(x) = B - Cx^{D}.

The length of the nose cone is 100 ft.

The base of the nose cone has a radius 20 ft.

The shape exponent D is 4, but as a first approximation, the designers
use D = 1 in order to simplify the mathematics.

The density of the nose cone varies linearly from 1 slug/ft^{3}
at the base to 2 slug/ft^{3} at the tip.

As a first approximation, assume the nose code
is a simple cone shape (D = 1 in the above equation). Where is the
centroid, center
of
mass,
and
center
of
gravity?

Approach

Use symmetry to determine the y and z locations of the centroid,
center of mass, and center of gravity.

Solve for the x location by integrating the appropriate quantities
from the base to the tip of the nose cone.