a couple of physics questions i truly don't understand...?

1. I believe you are correct. Part of the centripetal force is parallel to the bank (and driving the car out). Part of it is perpendicular to the bank (and contributing to the normal force). Gravity is also working in two directions (pulling the car down the bank, contributing to normal force). Friction (if present) is pulling the car down the bank. Question: does uphill component of centripetal force match downhill component of gravity? (Probably not.) If not, what's the difference?

2. Well, according to what I stated above, "perfectly banked" means

component of centripetal accel uphill = component of gravity downhill, or

v²/r • cosΘ = g•sinΘ

(19.4m/s)² / (80m•9.8m/s²) = 0.48 = sinΘ / cosΘ

I've never been good w/trig functions, and I don't know a rigorous way to solve this. A little trial and error gives Θ = 25.6º

At 25m/s, centripetal accel:

v²/r = (25m/s)²/80m = 7.8 m/s²

normal accel = 7.8m/s²•sin25.6º = 3.38 m/s²

uphill accel = 7.8m/s²•cos25.6º = 7.05 m/s²

gravity:

normal accel = 9.8m/s²•cos25.6º = 8.84 m/s²

downhill accel = 9.8m/s²•sin25.6º = 4.23 m/s²

net up-/down-hill accel = 2.82 m/s² UP

net normal accel = 12.22 m/s²

therefore the required µ = 2.82/12.22 = 0.23

the way I see it.

(I didn't bother multiplying by mass to get forces. Why bother?)