A sector of a circle has a perimeter of 7cm and an area of 3cm^2. Find all possible radii.
p=7
2r+s=7 K=1/2rs
K=3cm^2 K=1/2r(7-2r)
K=7/2r-r^2
s=7-2r
3=7/2r-r^2
-3 -3 -r^2+7/2r-3=0
/-1
r^2-7/2r+3=0
r= -3/2 r= -2
The formula we use for this problem is K=1/2rs, where r is the radius and s is the arc length. 7cm is to perimeter and 3cm^2 is the area. First you plug the 7cm into to perimeter formula and you get 2r+s=7. then you solve for s and you get s=7-2r. Then we plug the 7-2r into the K=1/2rs formula and we get K=7/2r-r^2. next, you plug your 3 into the formula as K because 3 is your area. You then subtract 3 from both sides and you get -r^2-7/2r-3=0. then you divide everything by one half. You end up with r^2-7/2r+3=0. You use the quadratic formula to solve for r because it is not factorable. The quadratic formula is b plus or minus the square root of negative b minus 4 times a times c, all divided by 2a. Once you plug in all the numbers and solve the equation, you will get r= -3/2 and r= -2.
