Answer
$\dfrac{-1}{(12\sqrt 3)} rad/s$
Work Step by Step
We need to use the chain rule.
$\dfrac{dA}{dt}=\dfrac{\partial A}{\partial x}\dfrac{dx}{ dt}+\dfrac{\partial A}{\partial y}\dfrac{dy}{ dt}+\dfrac{\partial A}{\partial \theta}\dfrac{d \theta}{ dt}$
Re-write as: $\dfrac{d \theta}{ dt}=-\dfrac{(\dfrac{\partial A}{\partial x}) \times (\dfrac{dx}{ dt})+(\dfrac{\partial A}{\partial y}) \times (\dfrac{dy}{ dt})}{(\dfrac{\partial A}{\partial \theta})}$
$=-\dfrac{y \sin \theta \times (\dfrac{dx}{dt})+(x \sin \theta) \times (\dfrac{dy}{ dt})}{(xy \cos \theta)}$
$=-\dfrac{(30) \times (0.5) \times (3)+(20) \times (0.5)(-2)}{(3) \times (20)(\sqrt 3/2)} $
Hence, we get $\dfrac{d \theta}{ dt}=\dfrac{-1}{(12\sqrt 3)} rad/s$
