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