The following diagram shows a sector of a circle where \({\rm{A\hat OB}} = x\) radians and the length of the \({\text{arc AB}} = \frac{2}{x}{\text{ cm}}\).

Given that the area of the sector is \(16{\text{ c}}{{\text{m}}^2}\), find the length of the arc \(AB\).

\({\text{arc length}} = \frac{2}{x} = rx\;\;\;\left( { \Rightarrow r = \frac{2}{{{x^2}}}} \right)\)     M1

\(16 = \frac{1}{2}{\left( {\frac{2}{{{x^2}}}} \right)^2}x\;\;\;\left( { \Rightarrow \frac{2}{{{x^3}}} = 16} \right)\)     M1

Note:     Award M1s for attempts at the use of arc-length and sector-area formulae.

\(x = \frac{1}{2}\)     A1

\({\text{arc length}} = {\text{4 (cm)}}\)     A1

[4 marks]