Hello!
Let's break this question down, answer by answer, to find the best option:
Federal government financing is an essential key in the survival of the interstate rail system.
(a) is an essential key in
There are two issues with this one. First, saying "an essential key" suggests there are other essential keys to keeping the interstate rail system going, but never explains what the others are. Also, it's better to say "for the survival..." than "in the survival..." in this case.
(b) is an essential key for
This one fixes the in/for problem, but still tells us that there are other "essential keys," rather than there being only one (which is what I think the writer is trying to say).
(c) is essential for
This is the best option. It fixes the in/for problem, and it drops the "key" to make it clearer that government funding alone is essential to keeping the rail system operating.
(d) is essential in
Again, this one correctly gets rid of the "key" problem, but still uses "in the survival" instead of the much clearer "for the survival."
(e) is essential
Read the sentence with this answer plugged in, and you'll see that it's missing something - not using the word "for" makes this clunky and not make sense.
I hope this helps! I'm available if you'd like any follow up.
