Things your Physics book has wrong (probably).
Oh, your professor learned from the same book.

Carnot Cycle solution? The Carnot Cycle and CarnotWork Ratio have nothing whatsoever to do with each other.
And, .... wait for it ....
The Carnot Cycle is the only known engine cycle in which the CarnotWork Ratio is not even part of the solution to the amount of work it does!
. . . Avast: Thar be the dredded eenty grails herein.

Reversible Herring Lord Kelvin's choice of words misleads text 'skimmers'.
An unintended red herring for the casual reader.
Reversibility is irrelevant.

Carnot Said Of all the stuff with Carnot's name glued on, which did he actually author?

No Reservoirs The only thing a Cold Reservoir could do is lose energy that could be turned into work.

Why Pressure-Volume Why Pressure-Volume graphs are used, what they mean, and how to use Heat drop to calculate Work.

Finding Gamma & Beta If your book still has Heat Capacity Ratio, the methodology printed is likely nonsense.