Hint Problem 1: Try and factor out the 3^x term and get a quadratic. For problem 3^(2x) -5(3^[x])=-6 use a substitution y = 3^x and you obtain y^2 - 5y + 6 = 0 and you can use the
quadratic equation to find a solution.
Hint Problem 2: Consider differentiating the function with respect to time.
Hint Problem 3: You used the sum of a geometric series and in R I got the following output:
For the rate of growth you have been given the information and depending on how you want to specify the variable, it can either be r = 1 + 0.0035 or r = 0.0035. In a geometric series formulation its r = 0.0035 and in another formulation its 1.0035.
Hint Problem 4: Assuming the problem is dealing with discrete differentials (geometric series) then consider the half life as the rate at which something decays every cycle (i.e. per period). You said that that the mass loss is 2.5% so this corresponds to the radioactive decay since the mass is proportional the atomic weight which represents a configuration of elementary particles (including protons, neutrons, and electrons). (I should point out, they are looking for unknowns currently that may appear in the LHC and other planned similar experiments).
Also please note, if you specify continuous rates of change you need to use the exponential formulation and if its discrete you need to use the series formulation (geometric series). If you need to use continuous then take a look at the exponential.