Optimizing Cobbs-Douglas given output
The problem is basically to derive the cost function of a firm. What you need to do is minimize costs given the output. That is
$$
\min\limits_{K,L} w \cdot L + r \cdot K
$$
subject to
$$
K^{\frac{1}{4}} \cdot L^{\frac{3}{4}} = Q = 120.
$$
You can solve this using Lagrange or the implicit function theorem (similar application as the MRS).