The formula for the area of a circle is #pir^2#. The radius of this circle is 3 feet. The area of the circle is #3^2pi#, which is equal to #9pi# ft#^2#. Since the angle of the arc that is shaded is 50 degrees, it is #50/360# of the whole circle. The area of that specific segment plus the area of the isosceles triangle is the area of a sector of the circle. The latter would be the total area of the circle multiplied by the portion of the circle that segment takes up.
The area is then #9pi * 50/360#, which is #5/4pi# ft#^2~~ 3.93 " ft"^2# .
To find the area of the isosceles triangle, note that its base is #2 times 3" ft" times sin(50^circ)/2#, while the height is #3" ft" times cos (50^circ)/2#. Thus, the area is
#1/2 times "base" times "height" = 1/2 times 2 times 3" ft" times sin(50^circ)/2 times 3" ft" times cos (50^circ)/2 = 4.5" ft"^2times sin50^circ ~~ 3.45" ft"^2#
Thus, the area of the shaded part is
#3.93 " ft"^2- 3.45" ft"^2 = 0.48" ft"^2 ~~ 0.5" ft"^2#(to the nearest tenth)
