Hello! I am given the following Producer's Problem in the Long Run and Short Run and have started to try and work it out. I wanted to make sure I am on the right track and have put my current solutions in bold to each problem.

Q = f(L,K) = 1/10 * L^0.5 * K ^0.5 where Q = Output, L is labor input, K is capital input. Input prices are w = 1 and r = 4 per unit of labor and capital respectively.

a) What is the firm's long-run cost as a function of output, C(Q;1,4)? I got C(Q;1,4) = 100Q/K+4K

b) What are the firm's long-run marginal cost and average cost as functions of output, MC(Q;1,4) and AC(Q;1,4), respectively? I got MC(Q; 1,4) = 100/K and AC(Q;1,4) = 100/K + 4K/Q

c) If the short-run capital is fixed at level K-bar = 10. What are the firm's short-run total cost, marginal cost, and average cost as functions of output? For STC(Q;1,4) = 10Q+40, SMC(Q;1,4) = 10 and SAC = 10+40Q