[MetaCRS] mercator scale calculation q

[Mark Amend]

Hi there-

This might not be an appropriate post for this list, but I'm struggling 
to find a solution.  Any ideas would be greatly appreciated.

I have a Mercator chart I'm building, at a scale of 1:5000 (Natural 
scale, at the equator, or in my case he projection has a defined 
standard parallel of 31N, where the scale factor is 1.00).  And yes, I 
have to use this projection.  ;-)

The chart, however, is located north of 31N, somewhere around 34N in the 
middle of the chart (for clarity of my question, let's just say it is 
34N).  I have been asked what is the True Scale of my chart at the 
mid-latitude of that chart.  So, it will not be exactly 1:5000 but 
something ± that, like 1:5043.1 or something (hypothetically speaking).  
I need to find an answer.

How do I calculate this "true scale" at the mid-latitude of my chart?  
It seems that Mercator scale is defined in various references as the 
secant of the latitude.  If that is the case, then ACOS(34 degrees? 
radians?) should give 1.xxx or so,  right?   But the scale for the 
latitude of interest (center of chart) is really 34 - 31 = 3,  ..if 31 
is where the scale factor is 1.  Right?  So it should be ACOS(3 
degrees)?  I then multiply 5000 by this number.  Correct?

Am I anywhere close to the solution?
Thanks,
Mark Amend