Regarding area sector and arc length

Question

The rear windshield wiper blade on a car has a length of 8  inches. The blade is mounted on a 11  inch arm, 3  inches from the pivot point. If the wiper turns through an angle of 150  degrees, how much area is swept clean?

Answer 1

The area of a circular sector is (1/2)r²θ, where r is the radius and θ is the measure of the central angle of the sector, measured in radians.  

 

θ = 150° = (150)(π/180) = 5π/6

 

Area cleaned by the blade =

 

 

 (Area of sector swept out by the entire arm) - (Area of the sector not                                                                       touched by the blade)

 

 =  (Area of sector with radius 11") - (Area of sector with radius 3")

 

 = (1/2)(11²)(5π/6) - (1/2)(3²)(5π/6)

 

 = (1/2)(11²-3²)(5π/6)

 

 ≈  146.6 in²