Minimize, solve block and function

I'm not sure how to formulate this in MC.  I want to have a function calculate area.  However, this area calculation has a minimize procedure built in  to find the angle that minimizes the area.  So, a flow area is a function of valve stroke and this angle that results from the stroke.  Easy enough when I define stroke length:

But when I want everything as a function of stroke I can't get anything to work.  I'm after something like this, but making things of a function of Stroke and Beta causes problems. 

3.1 files attached

Here's the paper if anyone anyone is interested or wants to visualize what I'm trying to do.

Calculating Throttling Area | Pneumatics content from Machine Design

Thanks,