The concentration of a specialized catalyst in a chemical reaction, CCC, is modeled by the function C(t)=(3+αt)4C(t) = (3 + \alpha t)^4C(t)=(3+αt)4 for small values of time t>0t > 0t>0, where α\alphaα is a non-zero constant.
Given that the coefficient of the t2t^2t2 term in the binomial expansion of (3+αt)4(3 + \alpha t)^4(3+αt)4 is 135013501350, find the possible values of α\alphaα.
A related adjusted concentration function, G(t)G(t)G(t), is defined by the product
G(t)=(127+2t3)(3+αt)4 G(t) = \left( \frac{1}{27} + \frac{2}{t^3} \right) (3 + \alpha t)^4 G(t)=(271+t32)(3+αt)4Using your values from part (a), determine the possible values for the term independent of ttt in the binomial expansion of G(t)G(t)G(t).
