3.22: Two sectors cut from a circle with variable inner radius

0606/11/O/N/22 — Question 7 · 6 marks

The diagram shows a circle with centre

O

and radius

r

OAB

and

OCD

are sectors of a circle with centre

O

and radius

x

, where

0 < x < r

. Angle

AOB =

angle

COD = \theta

radians, where

0 < \theta < \pi

(a)

Find, in terms of

r

x

and

\theta

, the perimeter of the shaded region.

[3]

(b)

Find, in terms of

r

x

and

\theta

, the area of the shaded region.

[1]

(c)

It is given that

x

can vary and that

r

and

\theta

are constant.

Write down the least possible area of the shaded region in terms of

r

and

\theta

[2]