PROBLEM $1$: $(10$ pts$)$

An artist wants to create an outdoor art installation with the shape shown

from $7-$gauge galvanized steel sheet. To design an appropriate support

structure, the center of gravity and moments of inertia with respect to

specified axes must be found. Since the object has a uniform thickness

and a constant density, it is only necessary to locate the centroid and to

determine the area moments of inertia with respect to the axes.

Calculate the following values for the area:

Horizontal location of the centroid $\stackrel{}{x}$

Vertical location of the centroid $\frac{?}{\text{b}\text{a}\text{r}}(y)$

Area moment of inertia about a horizontal axis passing through the

point $x=1m,y=3m.$

Area moment of inertia about a vertical axis passing through the

point $x=1m,y=3m.$

Polar moment of inertia about the perpendicular axis passing through the

point $x=1m,y=3m.$