If the area of a sector is 100 ft^2 and the radius of the circle is 22 ft what is the central angle measure for that sector and
what is the length of the arc
1 answer:
Answer:
![\displaystyle \theta=0.413\ rad](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ctheta%3D0.413%5C%20rad)
<em>L= 9.086 feet</em>
Step-by-step explanation:
<u>Area of a Circular Sector</u>
Given a circle of radius r, the area of a circular sector defined by a central angle θ (in radians) is given by
![\displaystyle A=\frac{1}{2}r^2\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20A%3D%5Cfrac%7B1%7D%7B2%7Dr%5E2%5Ctheta)
And the length of the arc is:
![L=\theta r](https://tex.z-dn.net/?f=L%3D%5Ctheta%20r)
We know the area of the sector is 100 square feet and the radius is r=33 ft, thus:
![\displaystyle 100=\frac{1}{2}r^2\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20100%3D%5Cfrac%7B1%7D%7B2%7Dr%5E2%5Ctheta)
Solving for θ:
![\displaystyle \theta=\frac{200}{r^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ctheta%3D%5Cfrac%7B200%7D%7Br%5E2%7D)
![\displaystyle \theta=\frac{200}{22^2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ctheta%3D%5Cfrac%7B200%7D%7B22%5E2%7D)
![\displaystyle \theta=0.413\ rad](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Ctheta%3D0.413%5C%20rad)
The arc length is:
![L=0.413* 22](https://tex.z-dn.net/?f=L%3D0.413%2A%2022)
L= 9.086 feet
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