Question

A sector of a circle of radius rm is bounded by an arc AB and by two radii OA and OB at an angle beta rad.Given that the perimeter of a sector is 18m and that the area of the sector is 8 metres square calculate the numerical values of r and beta.

Answer 1

Given: Radius of circle is r meter, perimeter of sector is 18 m and area of the sector is 8 m^2.

To find: The value of r and β.

Solution:

• Now we have given the radius of circle as r meter. Consider length of are BC as l, then:

                 OA + 0B + AB = 18 ...............(given)

                 r + r + l = 18

• Now we have angle AOB as β, so length will be rβ, so:

                 2r + rβ = 18 ...............(i)

• Now we have given area of sector as 8 m^2, so:

                 1/2 x r^2 x β = 8

                 β = 16/r^2 ................(ii)

• Putting (ii) in (i), we get:

                 2r + r(16/r^2) = 18

                 2r + 16/r = 18

                 2r^2 + 16 = 18r

                 2r^2 - 18r + 16 = 0

                 r^2 - 9r + 8 = 0

• Now by middle term splitting, we get:

                 (r-8)(r-1) = 0

                 r = 8,1

• Now consider r = 1, putting it in (ii), we get:

                 β = 16/1 = 16

• Now consider r = 8, putting it in (ii), we get:

                 β = 16/8^2 = 16/64 = 1/4

• So (r,β) = (1,16) and (8,1/4)

Answer:

               So the values of  (r,β) are (1,16) and (8,1/4)