A $($very thin$)1"x1"$ square tile is painted purple. The paint mass density increases steadily from one corner

to the opposite corner starting at $0o\frac{z}{s}q.$ inch and ending with $1oz/$sq inch. Where is the center of mass of

the tile? Suppose the tile is balanced on a pencil passing under the center of mass. Compute the moment of

inertia of the tile about an axis along the pencil for any possible such placement of the pencil. Which pencil

placement angle maximizes the moment of inertia?