.
Given a circle with sector AOB, centre O, radius r cm, where the acute angle AOB is θ radians. Given
that the perimeter of the sector is 14 cm and the area of the sector is 10 cm^2, evaluate r and θ
~~~~~~~~~~~~~~~~~~~~
The arc length is ; the sector area is .
Therefore, the equations for the problem are
2r + = 14 cm (the perimeter) (1)
= 10 cm^2 (the area) (2)
To find r, multiply equation (1) by r. You will get
+ = 14r. (3)
In (3), replace by 20, based on (2). You will get
+ 20 = 14r,
or
- 14r + 20 = 0,
- 7r + 10 = 0,
(r-5)*(r-2) = 0,
so the roots of (3) are r= 2 and r= 5.
If r = 2, then from (2) = = 5 radians.
If r = 5, then from (2) = = 0.8 radians.
Thus the problem has two solutions.
One solution is r= 2 cm, = 5 radians.
Other solution is r= 5 cm, = 0.8 radians.
Solved.