Answer:
The area of the associated sector is [tex]\frac{25}{24}\pi \ in^{2}[/tex]
Step-by-step explanation:
step 1
Find the radius of the circle
we know that
The circumference of a circle is equal to
[tex]C=2\pi r[/tex]
we have
[tex]C=5\pi\ in[/tex]
substitute and solve for r
[tex]5\pi=2\pi r[/tex]
[tex]r=2.5\ in[/tex]
step 2
Find the area of the circle
we know that
The area of the circle is equal to
[tex]A=\pi r^{2}[/tex]
we have
[tex]r=2.5\ in[/tex]
substitute
[tex]A=\pi (2.5^{2})=6.25\pi\ in^{2}[/tex]
step 3
Find the area of the associated sector
we know that
[tex]2\pi\ radians[/tex] subtends the complete circle of area [tex]6.25\pi\ in^{2}[/tex]
so
by proportion
Find the area of a sector with a central angle of [tex]\pi/3\ radians[/tex]
[tex]\frac{6.25\pi }{2\pi} =\frac{x}{\pi/3}\\x=6.25*(\pi/3)/2\\ \\x=\frac{25}{24}\pi \ in^{2}[/tex]
