Answer
$\dfrac{-1}{(12\sqrt 3)} rad/s$
Work Step by Step
Apply the chain rule:
$\dfrac{dA}{dt}=(\dfrac{\partial A}{\partial x})(\dfrac{dx}{ dt})+(\dfrac{\partial A}{\partial y})(\dfrac{dy}{ dt})+(\dfrac{\partial A}{\partial \theta})(\dfrac{d \theta}{ dt})$
This can be rearranged as:
$\dfrac{d \theta}{ dt}=-\dfrac{(\dfrac{\partial A}{\partial x})(\dfrac{dx}{ dt})+(\dfrac{\partial A}{\partial y})(\dfrac{dy}{ dt})}{(\dfrac{\partial A}{\partial \theta})}$
or, $\dfrac{d \theta}{ dt}=-\dfrac{y \sin \theta (\dfrac{dx}{dt})+(x \sin \theta)(\dfrac{dy}{ dt})}{(xy \cos \theta)}$
or, $\dfrac{d \theta}{ dt}=-\dfrac{(30) (0.5)(3)+(20)(0.5)(-2)}{(3)(20)(\sqrt 3/2)} $
or, $=\dfrac{-1}{(12\sqrt 3)} rad/s$
