I will give you some hints...but you need to work out the answers yourself in order for this to a learning exercise.

a) is a related rate problem. The area of the plate is πr^{2}. To find the related rates, differentiate the function

A=πr^{2} with respect to t.

b) is a maximum problem: differentiate the function for R with respect to θ and set it = to 0 to find the maximum.

c) is also a maximum problem: differentiate c(x) with respect to x and set it = 0 (same kind of problem as b)