Answer:
The first option is the correct one, the area of the shaded portion of the circle is
[/tex](5 \pi -11.6)ft^2[/tex]
Step-by-step explanation:
Let us first consider the triangle + the shadow.
The full area of the circle is the radius squared times pi, so
A=[tex](5 ft)^2 \cdot \pi \\25 ft^2 \cdot \pi[/tex]
Since [tex]\frac{72^{\circ}}{360^{\circ}}=\frac{1}{5}[/tex], the area of the triangle + the shaded area is one fifth of the area of the whole circle, thus
[tex]A_1=\frac{1}{5}25 ft^2 \cdot \pi\\ =5 ft^2 \cdot \pi[/tex]
If we want to know the area of the shaded part of the circle, we must subtract the area of the triangle from [tex]A_1[/tex].
The area of the triangle is given by
[tex]A_{triangle}=\frac{1}{2}\cdot (2.9+2.9)ft \cdot 4 ft\\= 11.6 ft^2[/tex]
Thus the area of the shaded portion of the circle is
[tex]A_1-A_{triangle}=5 \pi ft^2-11.6ft^2\\= (5 \pi -11.6)ft^2[/tex]
