A wire of length 20 ~cm is bent in the form of a sector of a circle. The maximum area that can be enclosed by the wire is

Step-by-step solution

Given that length of the wire P =20 ~cm Then, P = diameter + arc length aligned 20 &=2 r+S S &=20-2 r S &=2(10-r) ...(i) aligned Also, know that area of semicircle aligned & A = 1 2 r ² ...(ii) & A = 1 2 ( r )( r ) & Angle = Arc Radius & = S r aligned S=r for straight length of wire A = 1 2 ~S r ...(iii) From Eq. (i) A= 1 2 2(10-r) r A=10 r-r² ...(iv) Now, dA dr =10-2 r For max or min area of enclosed by wire dA dr =0 10-2 r =0 r =5 Then, from Eq. (iv) aligned &A=10(5)-(5)² &A=50-25 &A=25 sq cm aligned