Determining cost from production function, wage, and rental rate So I have a production function $Q=2K + 20L^{1/2}$ and I suppose the wage is $w=5$ and the rental rate is $r=9$.  I want to find the long-run cost of production, which I know is constrained by $\frac{MP_{L}}{MP_{K}}=w/r$ and therefore

$$\frac{10L^{-1/2}}{2} = 5/9$$

Which implies $L=81$.  However, I don't see how to find my quantity of capital, $K$.  Any help would be appreciated.
What's tricky is that the problem has a kink. Below a certain production value no capital is used. The demand functions for $k$ and $l$ are as follows:

If $Q \leq 180$
$$L(Q) = (Q/20)^2$$
$$K(Q) = 0$$

If $Q &gt; 180$
$$L(Q) = 81$$
$$K(Q) = (Q - 180) / 2$$

Which you can get from thinking about what happens to the marginal cost of producing with labor just below and above $L=81$ (goes from below 4.5 to above 4.5) and comparing it with the the cost of producing a marginal unit with capital (which is always 4.5).

Why the split at $Q = 180 / L = 81 $?
 $$\frac{MP_{L}}{MP_{K}}=w/r \leftrightarrow  \frac{10L^{-1/2}}{2} = 5/9 \rightarrow L = 81$$
Note that $ Q(k=0,L=81) = 180$