Answer:
[tex](5\pi-11.6)\ ft^{2}[/tex]
Step-by-step explanation:
we know that
The area of the shaded region is equal to the area of the sector minus the area of the triangle
step 1
Find the area of the circle
the area of the circle is equal to
[tex]A=\pi r^{2}[/tex]
we have
[tex]r=5\ ft[/tex]
substitute
[tex]A=\pi (5)^{2}[/tex]
[tex]A=25\pi\ ft^{2}[/tex]
step 2
Find the area of the sector
we know that
The area of the circle subtends a central angle of 360 degrees
so
by proportion find out the area of a sector by a central angle of 72 degrees
[tex]\frac{25\pi}{360}=\frac{x}{72}\\ \\x=72*25\pi /360\\ \\x=5\pi\ ft^{2}[/tex]
step 3
Find the area of triangle
The area of the triangle is equal to
[tex]A=\frac{1}{2}(2.9+2.9)(4)= 11.6\ ft^{2}[/tex]
step 4
Find the area of the shaded region
Subtract the area of the triangle from the area of the sector
[tex](5\pi-11.6)\ ft^{2}[/tex]
