Please Help? A circle has an arc length of 5Ο€ in. The central angle for this arc measures Ο€/3 radians. What is the area of the associated sector?

Accepted Solution

Answer:The area of the associated sector is [tex]\frac{25}{24}\pi \ in^{2}[/tex] Β  Step-by-step explanation:step 1Find the radius of the circlewe know thatThe 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 2Find the area of the circlewe know thatThe 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 3Find the area of the associated sectorwe know that[tex]2\pi\ radians[/tex] subtends the complete circle of area [tex]6.25\pi\ in^{2}[/tex]soby 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]