1) what radius of a circle is required to inscribe an equilateral triangle with an area of 15.588 in2 and an altitude of 5.196 in? (round to nearest tenth)

We know that
 the distance from the centroid of the triangle to one of the vertices is the radius of the circle required to inscribe an equilateral triangle.

[distance centroid of the triangle to one of the vertices]=(2/3)*h
h=the altitude  of the equilateral triangle> 5.196 in
so
[distance centroid of the triangle to one of the vertices]=(2/3)*5.196
[distance centroid of the triangle to one of the vertices]=3.464 in> 3.5 in

the radius is equal to the distance of the centroid of the triangle to one of the vertices
hence
the radius is 3.5 in

the answer is
the radius is 3.5 in

Agc and bgd are vertical angles