When is it preferable to use a plastic volumetric flask instead of a more accurate glass flask?

solution;

Typically plastic volumetric flasks are used when performing trace analysis and analyses may be absorbed OR desorbed from the glass.  

An examples is estimating [Na+] in aqueous solutions by a a process like ion-chromatography--if you acid wash your glass volumetric flasks then Na+ in the solution can be absorbed by the glass and then if you base wash your glass volumetric flask, then you may have excess Na+ leaching out into your solution. So to avoid leaching out or absorption of Na+ ions, you would want to use plastic volumetric flasks.  

The other instance you would use plastic vessels is measuring out solutions of strong bases like KOH and NaOH--glass slowly dissolves in solutions of bases--you will get potassium or sodium silicate in the end and glass is permanently etched by strong bases--so in these instances you would want to use plastic vessels  

As regards acids--the only problem you would have is with HF--glass would be etched/dissolved to form H₂SiF₆ and related soluble compounds  

they both have a y in them

it should have an internal chamber and two thermometers sticking out of the top. the internal chamber will also be closed off from the outside.