The perimeter of a sector is a constant. If its area is to be maximum, then the sectorial angle is

Step-by-step solution

Let r be the radius and be the sectorial angle. Perimeter of sector, aligned & k=2 r+ 180^ r & r= k 2+ 180^ aligned aligned Area of sector, A &= 360^ r ² &= 360^ [ ( k 2+ 180^ )² ] A &= k ² 360^ [ (2+ 180^ )⁻² ] aligned On differentiating w.r.t. ' ', we get aligned dA d =& k ² 360^ [1 (2+ 180^ )⁻²-2 (2+ 180^ )⁻³ . & .= k ² 360^ (2+ 180^ )⁻² [1- 180^ ) ] &= k ² 360^ (2+ 180^ )⁻³ [2- 180^ ] Put . dA d =0 2- 180^ ] = 2 180^ =2^ aligned At =2^ c , d² ~A d ² < 0 (maximum)