Eine Frau isst den Kase. Eine Frau --> Substantive Isst --> Verb Den Kase --> Object
- why "kase" is the object?
because its the target of the phrase. ex: if it was "the cheese is good"/"DER Kase is gut.", Kase could be the subject.
why "Den" and not "Der"?
- because it is following a object.
Short "ä" is easy: it's the same as German "short e", pretty much like English "short e" as in "pet".
German "long ä" is properly pronounced like "short ä" but longer -- i.e. the same vowel quality but a different vowel quantity.
Colloquially, many people pronounce "long ä" like "long e" (i.e. like French é), so Käse may sound like Keese or Räder (wheels) like Reeder (ship owners) or Bären (bears) like Beeren (berries). But that's not quite standard pronunciation.
You combined the indefinite article [for non-specific singular nouns] 'a' with the plural 'women'. 'Eine Frau' is singular: 'a woman'. This is the reason your answer wasn't accepted, though Duo's preference for 'lady' is odd.
'Woman' is the primary meaning of 'Frau', but it can also be translated as 'female', 'Mrs./Ms./Miss' or 'lady'. However, 'Dame' is the more appropriate -- and far more polite -- word for 'lady'.
Why doesn't "eats" work in this case?
What was your entire sentence?
Are you sure you had a translation exercise rather than a listening ("type what you hear") exercise?
Did you report your sentence as "my translation should be accepted"?
If you have a screenshot showing the question and your answer, it would be helpful if you upload it to a website somewhere (e.g. imgur) and told us the URL to the image.
Why this sentense is continuous ?
Both present simple and present continuous translations are accepted.
Without context, neither is more or less appropriate than the other -- German present tense is used for both things that happen now and for things that happen repeatedly.
Roughly: "the woman" is a woman who is known to the listener -- perhaps because you had spoken about her already.
"a woman" is a woman who has not been spoken about before. She is new to the conversation. Her identity is perhaps not important.
The distinction between eine Frau and die Frau in German is similar to that between "a woman" and "the woman" in English.
It has nothing to do with the subject, Mann or Frau. Rather, use of the indefinite article (in the English translation) depends on what they’re eating. “Apple” is a countable singular noun that starts with a vowel, and takes the article “an”. “Cheese” is an uncountable noun that starts with a consonant, and takes no indefinite article at all.
Nobody can see what you wrote, unless you show us, so references such as "this" or "my answer" is not helpful.
Show us your entire answer, please -- either by uploading your screenshot to a website somewhere and telling us the URL, or copy-and-pasting your entire answer into your comment.
my answer seems correct.
If you would like anyone to look at it, you will have to show it to us -- upload your screenshot to a website such as imgur and tell us the URL of the image.
Without a screenshot, there's not really anything anyone can do to guess what might have happened to you.