Coquus in culina manet, ut cenam magnam finire possit.

Because coquus is the subject of the main verb manet (stays/remains). In Latin, the subject of a finite verb is normally in the nominative case.

Culina is ablative because in with a stationary meaning (in/within, not into) takes the ablative:

• in culina = in the kitchen (location where he is staying)
  If it meant motion into the kitchen, Latin would use in
  • accusative (e.g., in culinam = into the kitchen).

Maneo, manēre can mean to remain/stay, and in many contexts it’s the natural meaning. English wait is often better expressed by exspectare or manere with an implied/explicit expectation, but here the point is that the cook remains in the kitchen in order to do something, so stays/remains fits best.

Here ut introduces a purpose clause: in order to. You can tell because:

1. It’s followed by a subjunctive verb (possit), and
2. The sentence describes an action (manet) done with a goal (so that he can finish…).

Because it is inside an ut purpose clause. Latin regularly uses:

• ut
  • subjunctive = so that / in order that
    So possit (present subjunctive of posse) is required by the construction.

The main verb manet is present tense, so the purpose clause typically uses the present subjunctive to show purpose in present time:

• manet … ut … possit = he stays … so that he can…
  If the main verb were past (e.g., mansit), you’d usually see the imperfect subjunctive: mansit … ut … posset.

Because cenam is the direct object of finire (to finish). Transitive verbs like finire take a direct object in the accusative.

Magnam agrees with cenam in case (accusative), number (singular), and gender (feminine), so it must be accusative too. It’s an adjective meaning big/large, so it modifies cenam:

• cenam magnam = a big dinner
  Word order in Latin is flexible; magnam can come before or after cenam without changing the basic meaning.

Yes. Cena can mean a meal, but it very commonly refers specifically to dinner (especially the main meal of the day in Roman context). The context often decides, and many textbooks gloss it as dinner by default.

Finire is the present active infinitive. It’s infinitive because it depends on posse:

• posse + infinitive = to be able to + verb
  So finire possit = can finish (literally, may be able to finish).

Both are possible. Latin often places the infinitive before the finite verb in such clauses, and it often places the subjunctive at the end of its clause. But ut cenam magnam possit finire would also be valid; it would just shift emphasis/style.

In many translations, so that he can is the most natural. The Latin subjunctive can sound like may in older/formal English, but in purpose clauses it usually communicates straightforward purpose: in order to be able to.

Yes, it could.

• ut … finiāt = so that he may finish (purpose focuses directly on finishing)
• ut … finire possit = so that he may be able to finish (purpose highlights ability/possibility—perhaps it’s difficult or time-sensitive)
  Latin often uses posse when the writer wants that nuance.

It belongs to the main clause:

• Main clause: Coquus in culina manet = The cook stays in the kitchen
• Purpose clause: ut cenam magnam finire possit = so that he can finish a big dinner
  Of course, logically the location supports the purpose, but grammatically it modifies manet.

Classical Latin has no definite/indefinite articles (the/a). Whether something is the or a is inferred from context. If Latin needs to be more specific, it can use demonstratives like hic/ille/is (this/that/he), but usually it doesn’t.