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The room sizes are rounded. They are not exatly the same. 2.9m and 2.7m I think.
So why bother posting them, then? Acoustics is science, and precision is important.
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But there is no "good ratio" with a room of 3mx5m with acceptable ceiling heights.
You seem to be going about it backwards: Start with the maximum possible ceiling height you can get, then see what other dimensions you can find for the remaining two, that are close to a good ratio. Tweak all three until you hit one of the best.
In your case, assuming that the height is 2.7m, and you have a 3m x 5m space to work with, then you do have some reasonable options:
H=2.50 m, W=3.00 m, L=3.75 m is exactly Louden's 5th best ratio (1 : 1.2 : 1.5)
H=2.60 m, W=2.97 m, L=3.62 m is exactly Sepmeyer's very best ratio (1 : 1.14 : 1.39)
And there are MANY other good ratios in that general region.
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I know the problem and I did a lot of calculation.
Looks like you didn't do the right calculations!

In any event, you didn't answer the original question with enough accuracy to be useful in helping out with your problem: What are the exact dimensions of your room? (Meaning the dimensions as it is right now, measured between the inner-leaf wall surfaces, not including the existing treatment).
- Stuart -