how do i find all possible radii using perimeter?

Question

A sector of a circle has perimeter 7 cm and area 3 cm^2. Find all possible radii.

Answer 1

The perimeter of a sector is 2r+s=7cm, where r=radius of circle and s is the arc length=rx, where x is the angle of the sector in radians. 360 degrees=2(pi) radians. s=rx, so 2r+rx=7=r(2+x). From this x=(7/r)-2 or (7-2r)/r. (Area of sector)/((pi)r^2)=x/(2(pi)). Therefore, 3/((pi)r^2)=(7-2r)/(2(pi)r); 3/r=(7-2r)/2; 6=7r-2r^2; 2r^2-7r+6=0; (r-2)(2r-3)=0. So r=2 or 3/2cm. (Since s=7-2r, s=3 or 4cm. The circumference is 4(pi) (12.57cm) or 3(pi) (9.42cm), so x/360=3/(4(pi)) or 4/(3(pi)), making x=85.94 (1.5(pi) radians) or 152.79 degrees (8/3(pi) radians.)